In this paper, the hyperoctahedral group algebra \(\mathscr{F}[\overrightarrow{S_{n}}]\) over a splitting field \(\mathscr{F}\) of wreath product \(\overrightarrow{S_{n}}\) with \(\text{char}(\mathscr{F})\nmid|\overrightarrow{S_{n}}|\), is considered and the unique idempotents corresponding to the four linear characters of the group \(\overrightarrow{S_{n}}\) are explored. Also, by establishing the minimum weights and dimensions, all group codes generated by the linear idempotents in the aforementioned group algebra are completely characterized for every \(n\). The nonlinear idempotents corresponding to nonlinear characters of \(\overrightarrow{S_{3}}\) are also obtained and various group codes in \(\mathscr{F}[\overrightarrow{S_{3}}]\) generated by linear and nonlinear idempotents are examined.
Coding theory originated long back in 1948 after Shannon [21] published a classic paper ‘‘A Mathematical Theory of Communication’’and was later explored by numerous researchers in different directions (see [4,10,7,6,8]). For instance, algebraic coding theory is a fast growing and fascinating topic which is popular for its mathematical structure and practical applications. Out of all mathematical techniques used to construct codes in algebraic coding theory, group algebra codes are of great importance and play a vital role in the error correction and detection for reliable communications while transferring information through noisy channels. In 1967, Berman [5] proved that cyclic codes and Reed Muller codes are ideals or subspaces in a group algebra \(\mathscr{F[G]}\) of a field \(\mathscr{F}\) and a finite group \(\mathscr{G}\) under consideration. For a deeper study of these types of codes one can refer to [2,5,16]. However, there exist some group algebra codes which are constructed from zero divisors and units in group rings [11], which do not necessarily form ideals in the algebraic structure. An important study regarding Abelian and non-Abelian group codes in a group algebra of a group over any field has been described in [3,15,19,20]. Further, the nature of group codes, whether Abelian or non-Abelian for \(|\mathscr{G}|<128\) with \(|\mathscr{G}|\notin\{24, 48, 54, 60, 64, 72, 96, 108, 120\}\) is explored in [19]. The aforementioned work motivated us to construct group codes in a group algebra of hyperoctahedral groups \(\overrightarrow{S_{n}}\). Furthermore, a study of group algebra codes over binary tetrahedral group can be found in [7].
Recall from [18] that a group algebra \(\mathscr{F[G]}=\big\{\sum\limits_{g\in \mathscr{G}} a_{g}g : a_{g}\in \mathscr{F}\big\}\) is the free \(\mathscr{F}\)-module over \(\mathscr{G}\) with the elements of group \(\mathscr{G}\) as an \(\mathscr{F}\)-basis for \(\mathscr{F[G]}\), where \(\mathscr{F}\) is a field and \(|\mathscr{G}|\) is finite. The addition and multiplication of elements in \(\mathscr{F[G]}\) are defined naturally by extending the operations in both \(\mathscr{F}\) and \(\mathscr{G}\) as: \(\sum\limits_{g\in \mathscr{G}} a_{g}g + \sum\limits_{g\in \mathscr{G}} b_{g}g=\sum\limits_{g\in \mathscr{G}} (a_{g}+b_{g})g\); \(\Big(\sum\limits_{g\in \mathscr{G}} a_{g}g\Big)\Big(\sum\limits_{h\in \mathscr{G}} b_{h}h\Big)=\sum\limits_{x\in \mathscr{G}} \Big(\sum\limits_{gh=x} a_{g}b_{h}\Big)x.\) Note that under these operations, \(\mathscr{F[G]}\) is an associative \(\mathscr{F}\)-algebra with identity \(1=1_{\mathscr{F}}1_{\mathscr{G}}\), where \(1_{\mathscr{F}}\) and \(1_{\mathscr{G}}\) are identity elements of \(\mathscr{F}\) and \(\mathscr{G}\) respectively. The scalar multiplication in \(\mathscr{F[G]}\) is defined by \(\alpha\Big(\sum\limits_{g\in \mathscr{G}} a_{g}g\Big)=\sum\limits_{g\in \mathscr{G}} (\alpha a_{g})g\), for \(\alpha\in \mathscr{F}\). Also, it has been found that an element \(g\in \mathscr{G}\) can be identified as an element \(1.g\) of \(\mathscr{F[G]}\) i.e., \(\mathscr{G}\subset \mathscr{F[G]}\) and hence the elements of \(\mathscr{G}\) constitute the coding basis for group codes when viewed as subspaces in \(\mathscr{F[G]}\).
Further, a group algebra code in \(\mathscr{F[G]}\) is defined as a one sided (left or right) ideal in \(\mathscr{F[G]}\), whereas if \(\mathscr{F[G]}\) is regarded as a vector space, then a code can be viewed as a subspace in \(\mathscr{F[G]}\). In addition, if \(\mathscr{G}\) is cyclic or Abelian, then a code in a group algebra \(\mathscr{F[G]}\) is a cyclic code or an Abelian code. On the other hand, if a code is a vector subspace of \(\mathscr{F[G]}\), then it is called a linear code. More importantly, if \(char(\mathscr{F})\nmid |\mathscr{G}|\), then \(\mathscr{F[G]}\) is semisimple. Thus, \(\mathscr{F[G]}\) is decomposable into a direct sum \(\mathscr{F[G]}=\oplus_{i} \mathscr{FG}_{e_{k}}\), where \(\mathscr{FG}_{e_{k}}\) is the minimal ideal generated by the idempotent \(e_{k}\) (see [12]). Suppose that \(\mathscr{E}=\{e_{i}\}_{i=1}^{s}\) is the set of idempotents in a semisimple group algebra \(\mathscr{F[G]}\), then any ideal \(\mathscr{I}\) in \(\mathscr{F[G]}\) is also expressible as a direct sum of minimal ideals in \(\{\mathscr{FG}_{e_{j}}\}_{e_{j}\in \mathscr{E}}\) which are generated by a single element \(e_{j}\in \mathscr{E}\), i.e., \(\mathscr{I}=\oplus_{x\in \mathscr{E}_{1}} \mathscr{FG}_{x}\), for some subset \(\mathscr{E}_{1}\) of \(\mathscr{E}\). Let \(\mathscr{I}\) be an ideal generated by a subset \(\mathscr{E^{'}}=\{e_{i}\}_{i=1}^{t}\) of the set of idempotents \(\mathscr{E}\subseteq \mathscr{F[G]}\). Then \(\mathscr{I}=\big\{u\in \mathscr{FG} : ue_{j}=0\ \text{for all}\ e_{j}\in \mathscr{E}\setminus \mathscr{E^{'}}\big\}\) and for simplicity \(\mathscr{I}\) is denoted by \(\mathscr{I}_{\mu}\), where \(\mu=\mathscr{E}\setminus \mathscr{E}^{'}\), the complement of \(\mathscr{E^{'}}\) in \(\mathscr{E}\). Moreover, the length of a group code \(\mathscr{I}_{\mu}\) in \(\mathscr{F[G]}\) is the order of the group \(\mathscr{G}\) and is denoted by \(n\). The weight \(wt\big(u\big)\) of an element \(u=\sum\limits_{g\in \mathscr{G}} a_{g}g\) is the order of the set which consists of all non-zero coefficients in the presentation of \(u\) with elements of \(\mathscr{G}\) as basis, i.e., \(wt\big(u\big)=|\big\{a_{g} : a_{g}\neq 0\big\}|\). Let \(\mathscr{I_{\mu}}\) be a group algebra code in \(\mathscr{F[G]}\) with its dimension as a subspace over \(\mathscr{F}\) is \(k\) and the minimum distance \(d\), where \(d=d\big(\mathscr{I_{\mu}}\big)=min\big\{wt\big(u\big) : 0\neq u\ \text{and}\ ue_{i}=0\ \text{for all}\ e_{i}\in\mu\big\}\). Then the group algebra code \(\mathscr{I_{\mu}}\) is called \((n,k,d)\) group code. Moreover, a linear \((n,k,d)\) group code over \(\mathscr{F}\) with \(d=n-k+1\) is called a maximum distance separable (MDS) group code.
Additionally, the algebraic structure of group algebra codes helps in detecting and correcting errors that may occur during reading or writing data. Further, algebraic properties in group algebra possess the capacity to design efficient decoding algorithms, reducing the time required for error correction. The use of group algebras lead to more efficient decoding processes compared to other coding schemes, particularly in terms of computational complexity. For instance, it follows from the above definition that producing a group code as an ideal in a group algebra involves the computations of product of elements with idempotents in the underlying group algebra. The complexity analysis of product of elements in group algebra has already been discussed in [1]. The results in [1] conclude that a computation of a product in group algebra apparently requires time \(O(n^{2})\) and the number of active multiplications can certainly be reduced to \(O(n^{\frac{3}{2}})\), where \(n\) is the order of the group in group algebra.
In fact, the study of group codes in a group algebra \(\mathscr{F[G]}\) depends on the choice of the field \(\mathscr{F}\) and the group \(\mathscr{G}\). The hyperoctahedral group \(\overrightarrow{S_{n}}\) is the wreath product of symmetric groups, i.e., \(\overrightarrow{S_{n}}=\mathbb{Z}_{2}\wr S_{n}\), the wreath product of a cyclic group of order \(2\) with the symmetric group over \(n\) symbols. This group can also be seen as the group of \(n\times n\) signed permutation matrices. The group \(\overrightarrow{S_{n}}\) of order \(2^{n}n!\) has its abstract presentation as a group generated by \(\sigma_{1}, \sigma_{2}, …, \sigma_{n},\) satisfying the relations \(\sigma_{i}^{2}=1\) \((i=1,2,…,n)\), \((\sigma_{i}\sigma_{i+1})^{3}=1\) \((i=1,2,…,n-2)\), \((\sigma_{n-1}\sigma_{n})^{4}=1\) and \(\sigma_{j}\sigma_{k}=\sigma_{k}\sigma_{j}\), \(\forall j\) and \(k\neq j\pm 1\).
Throughout this paper, \(\overrightarrow{S_{n}}\) represents the hyperoctahedral group of dimension \(n\) and \(\mathscr{F}\) is a splitting field of \(\overrightarrow{S_{n}}\) whose order is relatively prime to \(char(\mathscr{F})\). The main objective of this paper is to characterize completely the group codes generated by the linear idempotents in the group algebra \(\mathscr{F}[\overrightarrow{S_{n}}]\) corresponding to the linear characters of \(\overrightarrow{S_{n}}\), for every \(n\). Also, various group codes of length \(48\) generated by linear and nonlinear idempotents in \(\mathscr{F}[\overrightarrow{S_{3}}]\) are obtained in section 3.
The group \(\overrightarrow{S_{n}}\) has \(p_{b}(n)\) conjugacy classes, where \(p_{b}(n)\) represents the number of bi-partitions of a natural number \(n\) (see [17,9]). Consequently, the group \(\overrightarrow{S_{n}}\) has \(p_{b}(n)\) characters, out of which four are linear characters (say) \(\chi_{1},\chi_{2},\chi_{3},\chi_{4}\), as discussed in the upcoming remark, while all the remaining characters are nonlinear. Recall that a linear character of a group \(\mathscr{G}\) is a group homomorphism from \(\mathscr{G}\) to the multiplicative group \(\mathscr{F^{*}}\) of a field \(\mathscr{F}\). In fact, the linear characters of a group are the only non zero characters which are homomorphisms from that group to the multiplicative group of a field and the set of all linear characters of a group forms an Abelian group under point wise multiplication which is called the character group of that group.
Remark 2.1. [17] Let \(\sigma_{1}, \sigma_{2}, …, \sigma_{n}\) be the generators of the hyperoctahedral group \(\overrightarrow{S_{n}}\) of dimension \(n\geq 2\). The character group of \(\overrightarrow{S_{n}}\) is of order 4 i.e., \(\overrightarrow{S_{n}}\) has four linear characters \(\chi_{1},\chi_{2},\chi_{3},\chi_{4} : \overrightarrow{S_{n}}\longrightarrow\mathscr{F^{*}}\) defined by \(\chi_{1}(\sigma_{j})=\ 1, \ \forall \ j, 1\leq j\leq n\); \(\chi_{2}(\sigma_{j})=-1, \ \forall \ j, 1\leq j\leq n\); \[\chi_{3}(\sigma_{j})= \begin{cases} \ \ 1,\ \forall \ j\leq n-1,\\ -1,\ \text{for}\ j=n,\\ \end{cases}\] and \[\chi_{4}(\sigma_{j})= \begin{cases} -1,\ \forall \ j\leq n-1,\\ \ \ 1,\ \text{for}\ j=n.\\ \end{cases}\]
As we have gathered all the necessary information required to obtain the linear idempotents corresponding to the linear characters in \(\overrightarrow{S_{n}}\), so we list them in the following lemma by using Proposition 14.10 in [14], which states that: ‘‘If \(\chi\) is a character of \(\mathscr{F}[\mathscr{G}]\)-module, then the unique idempotent corresponding to the character \(\chi\) is given by \(e=\frac{1}{|\mathscr{G}|}\sum\limits_{g\in \mathscr{G}}^{} \chi(g^{-1})g\)’’. Moreover, an idempotent is termed as linear or nonlinear idempotent, if it corresponds to a linear or nonlinear character respectively.
Lemma 2.2. The linear idempotents in \(\mathscr{F}[\overrightarrow{S_{n}}]\) are given by \(e_{1}=\frac{1}{2^{n}n!} \sum\limits_{g\in \overrightarrow{S_{n}}} \chi_{1}(g^{-1})g=\frac{1}{2^{n}n!} \sum\limits_{g\in \overrightarrow{S_{n}}}g\); \(e_{2}=\frac{1}{2^{n}n!} \sum\limits_{g\in \overrightarrow{S_{n}}} \chi_{2}(g^{-1})g\); \(e_{3}=\frac{1}{2^{n}n!} \sum\limits_{g\in \overrightarrow{S_{n}}} \chi_{3}(g^{-1})g\) and \(e_{4}=\frac{1}{2^{n}n!} \sum\limits_{g\in \overrightarrow{S_{n}}} \chi_{4}(g^{-1})g\).
Further, we compute the product of an arbitrary element and the linear idempotents in \(\mathscr{F}[\overrightarrow{S_{n}}]\) which will be used to construct various group codes in \(\mathscr{F}[\overrightarrow{S_{n}}]\). For this, let \(u=\sum\limits_{g\in \overrightarrow{S_{n}}} \lambda_{g}g\) be an arbitrary element in \(\mathscr{F}[\overrightarrow{S_{n}}]\) with \(\lambda_{g}\in \mathscr{F}\). Then
\[\begin{aligned}
ue_{i} & =\Big(\sum\limits_{g\in \overrightarrow{S_{n}}}
\lambda_{g}g\Big)\Big(\frac{1}{2^{n}n!} \sum\limits_{h\in
\overrightarrow{S_{n}}} \chi_{i}(h^{-1})h\Big) \\
& = \frac{1}{2^{n}n!}\sum\limits_{x\in
\overrightarrow{S_{n}}}\Big(\sum\limits_{g,h\in
\overrightarrow{S_{n}},gh=x} \lambda_{g}\chi_{i}(h^{-1})\Big)x \\
& = \frac{1}{2^{n}n!}\sum\limits_{x\in
\overrightarrow{S_{n}}}\Big(\sum\limits_{g,h\in
\overrightarrow{S_{n}},gh=x}
\lambda_{g}\chi_{i}(h^{-1})\Big)\chi_{i}(xx^{-1})x,
\end{aligned}\] as for any character \(\chi\), \(\chi(xx^{-1})=\chi(e)=1\), where \(e\) is the identity element of \(\overrightarrow{S_{n}}\).
Further, since \(\chi_{i}\) is a linear
character and every linear character is a group homomorphism, we have
\(\chi_{i}(xx^{-1})=\chi_{i}(x)\chi_{i}(x^{-1})\),
therefore
\[\begin{aligned} ue_{i} & = \frac{1}{2^{n}n!}\sum\limits_{x\in \overrightarrow{S_{n}}}\Big(\sum\limits_{g,h\in \overrightarrow{S_{n}},gh=x} \lambda_{g}\chi_{i}(h^{-1})\Big)\chi_{i}(x)\chi_{i}(x^{-1})x. \end{aligned}\]
Note that \(\chi_{i}(g)\in\mathscr{F}\), for all \(g\in\overrightarrow{S_{n}}\) and \(i=1,2,3,4\), so \(\chi_{i}(x)\) can be distributed inside the summation, thus the above can be written as
\[\begin{aligned} ue_{i} & = \frac{1}{2^{n}n!}\sum\limits_{x\in \overrightarrow{S_{n}}}\Big(\sum\limits_{g,h\in \overrightarrow{S_{n}},gh=x} \lambda_{g}\chi_{i}(h^{-1})\chi_{i}(x)\Big)\chi_{i}(x^{-1})x. \end{aligned}\]
Again, \(\chi_{i}\) is a homomorphism, therefore
\[\begin{aligned} ue_{i} & = \frac{1}{2^{n}n!}\sum\limits_{x\in \overrightarrow{S_{n}}}\Big(\sum\limits_{g,h\in \overrightarrow{S_{n}},gh=x} \lambda_{g}\chi_{i}(xh^{-1})\Big)\chi_{i}(x^{-1})x \\ & = \frac{1}{2^{n}n!}\sum\limits_{x\in \overrightarrow{S_{n}}}\Big(\sum\limits_{g\in \overrightarrow{S_{n}}} \lambda_{g}\chi_{i}(g)\Big)\chi_{i}(x^{-1})x \\ & =\Big(\sum\limits_{g\in \overrightarrow{S_{n}}} \lambda_{g}\chi_{i}(g)\Big)\Big(\frac{1}{2^{n}n!} \sum\limits_{x\in \overrightarrow{S_{n}}} \chi_{i}(x^{-1})x\Big) \\ & =\Big(\sum\limits_{g\in \overrightarrow{S_{n}}} \lambda_{g}\chi_{i}(g)\Big)e_{i}. \end{aligned}\]
Thus, we summarize \[\ ue_{1} =\Big(\sum\limits_{g\in \overrightarrow{S_{n}}} \lambda_{g}\chi_{1}(g)\Big)e_{1}=\Big(\sum\limits_{g\in \overrightarrow{S_{n}}} \lambda_{g}\Big)e_{1},\label{pg11}\tag{1}\] \[ue_{2} =\Big(\sum\limits_{g\in \overrightarrow{S_{n}}} \lambda_{g}\chi_{2}(g)\Big)e_{2}, \label{pg12}\tag{2}\] \[ue_{3} =\Big(\sum\limits_{g\in \overrightarrow{S_{n}}} \lambda_{g}\chi_{3}(g)\Big)e_{3}, \label{pg13} \tag{3}\] \[ue_{4} =\Big(\sum\limits_{g\in \overrightarrow{S_{n}}} \lambda_{g}\chi_{4}(g)\Big)e_{4}.\label{pg14}%4 \tag{4}\]
Proposition 2.3. The system of four equations \(\sum\limits_{g\in \overrightarrow{S_{n}}} \lambda_{g}\chi_{i}(g)=0\), \(i=1,2,3,4\) is linearly independent.
Proof . Let \(AX=0\) be the matrix representation for the system of equations \(\sum\limits_{g\in \overrightarrow{S_{n}}} \lambda_{g}\chi_{i}(g)=0\) where \(i=1,2,3,4\). Clearly, the matrix \(A\) is a matrix of order \(4\times 2^{n}n!\) whose \(j^{th}\) column contains the coefficients of the \(j^{th}\) unknown \(\lambda_{j}\) in the given system of four equations for \(i=1,2,3,4\) and \(X\) is a \(2^{n}n!\times 1\) column matrix of \(2^{n}n!\) unknowns \(\lambda_{j}\), \(1\leq j\leq 2^{n}n!\). Note that the given system of equations is linearly independent if and only if the rank of the matrix \(A\) is 4. Equivalently, the matrix \(A\) has a non vanishing minor of order 4.
Next, we will construct a square sub-matrix of \(A\) of order 4 with non zero determinant. For this, first observe that each column of the matrix \(A\) corresponds to an element \(g\in\overrightarrow{S_{n}}\) which is for sure the column vector \(\begin{bmatrix} \chi_{1}(g) \\ \chi_{2}(g) \\ \chi_{3}(g) \\ \chi_{4}(g) \end{bmatrix}\), where \(\chi_{1}, \chi_{2}, \chi_{3}, \chi_{4}\) are linear characters of \(\overrightarrow{S_{n}}\). Now, consider a square sub matrix \(B\) of order 4 of the matrix \(A\), whose columns are the columns of the matrix \(A\) corresponding to four distinct elements \({\sigma_{1}}^{2}, \sigma_{1}, \sigma_{n}\) and \(\sigma_{1}\sigma_{n}\in\overrightarrow{S_{n}}\) is
\[B= \begin{bmatrix} \chi_{1}({\sigma_{1}}^{2}) & \chi_{1}(\sigma_{1}) & \chi_{1}(\sigma_{n}) & \chi_{1}(\sigma_{1}\sigma_{n}) \\ \chi_{2}({\sigma_{1}}^{2}) & \chi_{2}(\sigma_{1}) & \chi_{2}(\sigma_{n}) & \chi_{2}(\sigma_{1}\sigma_{n}) \\ \chi_{3}({\sigma_{1}}^{2}) & \chi_{3}(\sigma_{1}) & \chi_{3}(\sigma_{n}) & \chi_{3}(\sigma_{1}\sigma_{n}) \\ \chi_{4}({\sigma_{1}}^{2}) & \chi_{4}(\sigma_{1}) & \chi_{4}(\sigma_{n}) & \chi_{4}(\sigma_{1}\sigma_{n}) \end{bmatrix}.\]
Since \(\chi_{1}, \chi_{2}, \chi_{3}\) and \(\chi_{4}\) are linear characters of \(\overrightarrow{S_{n}}\), therefore the entries of the above matrix reduces to
\[B= \begin{bmatrix} \chi_{1}({\sigma_{1}})^{2} & \chi_{1}(\sigma_{1}) & \chi_{1}(\sigma_{n}) & \chi_{1}(\sigma_{1})\chi_{1}(\sigma_{n}) \\ \chi_{2}({\sigma_{1}})^{2} & \chi_{2}(\sigma_{1}) & \chi_{2}(\sigma_{n}) & \chi_{2}(\sigma_{1})\chi_{2}(\sigma_{n}) \\ \chi_{3}({\sigma_{1}})^{2} & \chi_{3}(\sigma_{1}) & \chi_{3}(\sigma_{n}) & \chi_{3}(\sigma_{1})\chi_{3}(\sigma_{n}) \\ \chi_{4}({\sigma_{1}})^{2} & \chi_{4}(\sigma_{1}) & \chi_{4}(\sigma_{n}) & \chi_{4}(\sigma_{1})\chi_{4}(\sigma_{n}) \end{bmatrix}.\]
Further, in view of Remark 2.1, the above matrix takes the form
\[B= \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & -1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & 1 & -1 \end{bmatrix} \ \text{and}\ |B|\neq 0.\] ◻
From the above proposition, it follows that the four equations \(\sum\limits_{g\in \overrightarrow{S_{n}}}
\lambda_{g}\chi_{i}(g)=0\), (for \(i=1,2,3,4\)) are linearly independent. It
is pertinent to notice that out of these four equations if we choose two
or three distinct equations at a time, again form a linearly independent
set, as a subset of a linearly independent set is linearly independent.
The above observation infers that the dimension of an ideal \(\mathscr{I}_{\mu}=\{u\in
\mathscr{F}[\overrightarrow{S_{n}}] : ux=0\ \text{for all}\
x\in\mu\}\) where \(\mu\subseteq\{e_{1}, e_{2}, e_{3},
e_{4}\}\), as a vector subspace of \(\mathscr{F}[\overrightarrow{S_{n}}]\) is
\(2^{n}n!-|\mu|\), because \(ue_{i}=0\) if and only if \(\sum\limits_{g\in \overrightarrow{S_{n}}}
\lambda_{g}\chi_{i}(g)=0\), for \(i=1,2,3,4\) and every equation imposes an
independent constraint.
The upcoming theorems provide us with various group codes in \(\mathscr{F}[\overrightarrow{S_{n}}]\).
Theorem 2.4. Let \(e_{1}\), \(e_{2}\), \(e_{3}\), \(e_{4}\) be linear idempotents in \(\mathscr{F}[\overrightarrow{S_{n}}]\). Then \(\mathscr{I}_{\{e_{i}\}}\) for \(i=1,2,3,4\), is \((2^{n}n!, 2^{n}n!-1, 2)\) MDS group code.
Proof . Assume \(u=\lambda g\), for \(g\in\overrightarrow{S_{n}}\) and \(0\neq\lambda\in \mathscr{F}\) so that \(wt(u)=1\). By using (1) to (4), we have \(ue_{i}=\pm\lambda\chi_{i}(g) e_{i}\neq 0\) for \(i=1,2,3,4\). This implies that \(u\notin \mathscr{I}_{\{e_{i}\}}\) and hence the minimum distance of \(\mathscr{I}_{\{e_{i}\}}\) given by \(d(\mathscr{I}_{\{e_{i}\}})= min\big\{wt\big(u\big) : 0\neq u\ \text{and}\ ue_{i}=0\}\) is \(\geq 2\). Now, for any choice of two distinct elements \(\sigma_{j}, \sigma_{k}\) from the subset \(\{\sigma_{1}, \sigma_{2}, …, \sigma_{n-1}\}\) of the set of generators of \(\overrightarrow{S_{n}}\), there exists an element \(u=\sigma_{j}-\sigma_{k}\in \mathscr{F}[\overrightarrow{S_{n}}]\) with weight \(2\) such that \(ue_{i}=(\chi_{i}(\sigma_{j})-\chi_{i}(\sigma_{k}))e_{i}\). Further, Remark 2.1 infers that \(ue_{i}=0\), as \(\chi_{i}(\sigma_{j})=\chi_{i}(\sigma_{k})\) for all \(j,k\in\{1,2,…,n-1\}\). This implies that \(u\in \mathscr{I}_{\{e_{i}\}}\) for every \(i\) and hence \(d(\mathscr{I}_{\{e_{i}\}})=2\). Moreover, as discussed above, the dimension of \(\mathscr{I}_{\{e_{i}\}}\) as a vector space over \(\mathscr{F}\) is \(2^{n}n!-1\). Hence \(\mathscr{I}_{\{e_{i}\}}\) is a \((2^{n}n!, 2^{n}n!-1, 2)\) MDS group code. ◻
Theorem 2.5. Let \(e_{1}\), \(e_{2}\), \(e_{3}\), \(e_{4}\) be linear idempotents in \(\mathscr{F}[\overrightarrow{S_{n}}]\) and \(i, j, k\) be three distinct elements of the set \(\{1, 2, 3, 4\}\). Then
\((i)\) \(\mathscr{I}_{\{e_{i}, e_{j}\}}\) is \((2^{n}n!, 2^{n}n!-2, 2)\) group code.
\((ii)\) \(\mathscr{I}_{\{e_{i}, e_{j}, e_{k}\}}\) is \((2^{n}n!, 2^{n}n!-3, 2)\) group code.
\((iii)\) \(\mathscr{I}_{\{e_{1}, e_{2}, e_{3}, e_{4}\}}\) is \((2^{n}n!, 2^{n}n!-4, 2)\) group code.
Proof. Assume that \(\mathscr{I}\) denotes the corresponding ideal in \((i)\), \((ii)\) and \((iii)\). First, it is observed from equation (1) to(4) that for \(i\in\{1,2,3,4\}\), \(ue_{i}=0\) if and only if \(\sum\limits_{g\in \overrightarrow{S_{n}}} \lambda_{g}\chi_{i}(g)=0\) is satisfied.
Now, in any of the parts \((i)\), \((ii)\) and \((iii)\), if we choose an element \(u=\lambda_{g} g\), for \(g\in\overrightarrow{S_{n}}\), \(0\neq\lambda_{g}\in \mathscr{F}\) with \(wt(u)=1\), then it follows from equation (1) to (4) that \(ue_{i}=0\) for \(i=1,2,3,4,\) if and only if \(\lambda_{g}=0\). This implies that \(u\notin \mathscr{I}\) and so \(d(\mathscr{I})\geq 2\). We claim that for any two distinct choices of generators \(\sigma_{s}, \sigma_{t}\) from a subset \(\{\sigma_{1}, \sigma_{2}, …, \sigma_{n-1}\}\) of the set of generators of \(\overrightarrow{S_{n}}\), there exists an element \(u=\sigma_{s}-\sigma_{t}\in \mathscr{F}[\overrightarrow{S_{n}}]\) such that \(u\in \mathscr{I}\). In order to show this, it is enough to prove that \(ue_{i}=0\), for each \(i=1, 2, 3, 4\). Clearly, for \(u=\sigma_{s}-\sigma_{t}\), we have \(\lambda_{\sigma_{s}}=1, \lambda_{\sigma_{t}}=-1\) and \(\lambda_{g}=0\), for all \(g\in\overrightarrow{S_{n}}\setminus\{\sigma_{s}, \sigma_{t}\}\). Therefore, by using (1) to (4), we have \(ue_{i}=(\chi_{i}(\sigma_{s})-\chi_{i}(\sigma_{t}))e_{i}.\) Again, by using Remark 2.1, we have \(ue_{i}=0\), as \(\chi_{i}(\sigma_{s})=\chi_{i}(\sigma_{t})\) for all \(s,t\in\{1,2,…,n-1\}\). Thus, \(d(\mathscr{I})=2\). Moreover, the dimension of ideals \(\mathscr{I}\) over \(\mathscr{F}\) in the case \((i)\), \((ii)\) or \((iii)\) is \(2^{n}n!-2\), \(2^{n}n!-3\) or \(2^{n}n!-4\) respectively and this finishes the proof. ◻
The group \(\overrightarrow{A_{n}}\) of order \(2^{n-1}n!\) can be defined as the kernel of the homomorphism \(\epsilon : \overrightarrow{S_{n}}\longrightarrow \{\pm1\}\). Also, it can be viewed as the wreath product of a cyclic group of order \(2\) with the alternating group \(A_{n}\) i.e., \(\overrightarrow{A_{n}}=\mathbb{Z}_{2}\wr A_{n}\). For \(n\geq 2\), \(\overrightarrow{A_{n}}\) has \(\frac{1}{2}p_{b}(n)\) or \(\frac{1}{2}p_{b}(n)+\frac{3}{2}p(n)\) characters depending on whether \(n\) is even or odd respectively (see [9]). There are two linear characters of \(\overrightarrow{A_{n}}\), namely, \({\chi_{1}}^{'}\) the trivial character and \({\chi_{2}}^{'}\), the restriction of the linear character \(\chi_{1}\) of \(\overrightarrow{S_{n}}\) as already discussed in Remark 2.1.
Finally, for \(\overrightarrow{A_{n}}\), using the same arguments with the necessary variations as in case of \(\overrightarrow{S_{n}}\), one can easily obtain the linear idempotents and the group codes generated by these idempotents in a group algebra \(\mathscr{F}[\overrightarrow{A_{n}}]\) as discussed in the following results, where \(\mathscr{F}\) is a splitting field of \(\overrightarrow{A_{n}}\) with \(char(\mathscr{F})\nmid|\overrightarrow{A_{n}}|\).
Lemma 3.1. The linear idempotents in \(\mathscr{F}[\overrightarrow{A_{n}}]\) are given by \(f_{1}=\frac{1}{2^{n-1}n!} \sum\limits_{g\in \overrightarrow{A_{n}}} {\chi_{1}}^{'}(g^{-1})\\g=\frac{1}{2^{n-1}n!} \sum\limits_{g\in \overrightarrow{A_{n}}} g\) and \(f_{2}=\frac{1}{2^{n-1}n!} \sum\limits_{g\in \overrightarrow{A_{n}}} {\chi_{2}}^{'}(g^{-1})g.\)
Theorem 3.2. Let \(f_{1}\), \(f_{2}\) be linear idempotents in \(\mathscr{F}[\overrightarrow{A_{n}}]\). Then
\((i)\) \(\mathscr{I}_{\{f_{i}\}}\) for \(i=1,2\), is \((2^{n-1}n!, 2^{n-1}n!-1, 2)\) MDS group code.
\((ii)\) \(\mathscr{I}_{\{f_{1}, f_{2}\}}\) is \((2^{n-1}n!, 2^{n-1}n!-2, 2)\) group code.
The hyperoctahedral group of dimension \(3\) is \(\overrightarrow{S_{3}}\) which has its abstract presentation as
\[\langle a, b, c : a^{2}=b^{2}=c^{2}=1, (ab)^{3}=1, (bc)^{4}=1, ab\neq ba, bc\neq cb, ac=ca\rangle.\]
Thus, the elements of \(\overrightarrow{S_{3}}\) as words, using
the letters \(a\), \(b\) and \(c\) can be listed in a set as
follows:
\(\overrightarrow{S_{3}}=\{1, a, b, aba, cbc,
acbac, bacbacb, ab, ba, acbc, cbac, abacbc, acbacb, bacbac, bcbacb, c,
bcb, abcba, ac, bacb,\)\( abacba, cbacbc, abcbacbc, bacbacbc, bcbc,
abcbac, abacbacb, bc, cb, abac, abcb, acba, bcba, abacbacbc,
abcbc, bacbc,\)\( bcbac, cbacb, abacbac, abcbacb, abc, acb, bac, cba, abacb,
bacba, acbacbc, bcbacbc\}\).
Moreover, \(\overrightarrow{S_{3}}\) has ten conjugacy classes which are given by
\[\begin{aligned} C_{1}&=\{1\},\\ C_{2}&=\{a, b, aba, cbc, acbac, bacbacb\},\\ C_{3}&=\{ab, ba, acbc, cbac, abacbc, acbacb, bacbac, bcbacb\},\\ C_{4}&=\{c, bcb, abcba\},\\ C_{5}&=\{ac, bacb, abacba, cbacbc, abcbacbc, bacbacbc\},\\ C_{6}&=\{bcbc, abcbac, abacbacb\}\\, C_{7}&=\{bc, cb, abac, abcb, acba, bcba\},\\ C_{8}&=\{abacbacbc\},\\ C_{9}&=\{abcbc, bacbc, bcbac, cbacb, abacbac, abcbacb\}, \\ C_{10}&=\{abc, acb, bac, cba, abacb, bacba, acbacbc, bcbacbc\}. \end{aligned}\]
Consequently, the group \(\overrightarrow{S_{3}}\) has ten characters, precisely, four linear characters \(\chi_{1},\chi_{2},\chi_{3},\chi_{4}\) and six nonlinear characters \(\chi_{5},\chi_{6},\chi_{7},\chi_{8},\chi_{9},\chi_{10}\), out of which the characters \(\chi_{5},\chi_{6}\) are of degree \(2\) and \(\chi_{7},\chi_{8},\chi_{9},\chi_{10}\) are of degree \(3\) discussed in Table 1.
| Conjugacy Classes | ||||||||||
| Characters | \(C_1\) | \(C_2\) | \(C_3\) | \(C_4\) | \(C_5\) | \(C_6\) | \(C_7\) | \(C_8\) | \(C_9\) | \(C_10\) |
| \(\chi_1\) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| \(\chi_2\) | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 |
| \(\chi_3\) | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 |
| \(\chi_4\) | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | – 1 | 1 |
| \(\chi_5\) | 2 | 0 | -1 | 2 | 0 | 2 | 0 | 2 | 0 | -1 |
| \(\chi_6\) | 2 | 0 | -1 | -2 | 0 | 2 | 0 | -2 | 0 | 1 |
| \(\chi_7\) | 3 | 1 | 0 | 1 | -1 | -1 | 1 | -3 | -1 | 0 |
| \(\chi_8\) | 3 | -1 | 0 | -1 | -1 | -1 | 1 | 3 | 1 | 0 |
| \(\chi_9\) | 3 | 1 | 0 | -1 | 1 | -1 | -1 | 3 | -1 | 0 |
| \(\chi_10\) | 3 | -1 | 0 | 1 | 1 | -1 | -1 | -3 | 1 | 0 |
Now onwards, for our own convenience, we denote the class sum of a conjugacy class \(C_{i}\) by \(\overline{C_{i}}\) and the \(i^{th}\) element of the set \(\overrightarrow{S_{3}}\) defined in the beginning of this section by \(g_{i}\). At this point, we are in a position to obtain the idempotents corresponding to the characters of \(\overrightarrow{S_{3}}\), so in view of Proposition 14.10 in [14], we list them in the following two lemmas.
Lemma 4.1. The linear idempotents in \(\mathscr{F}[\overrightarrow{S_{3}}]\) are given by
\(e_{1}=\frac{1}{48}[1+\overline{C_{2}}+\overline{C_{3}}+\overline{C_{4}}+\overline{C_{5}}+\overline{C_{6}}+\overline{C_{7}}+\overline{C_{8}}+\overline{C_{9}}+\overline{C_{10}}]\),
\(e_{2}=\frac{1}{48}[1-\overline{C_{2}}+\overline{C_{3}}-\overline{C_{4}}+\overline{C_{5}}+\overline{C_{6}}+\overline{C_{7}}-\overline{C_{8}}-\overline{C_{9}}-\overline{C_{10}}]\),
\(e_{3}=\frac{1}{48}[1+\overline{C_{2}}+\overline{C_{3}}-\overline{C_{4}}-\overline{C_{5}}+\overline{C_{6}}-\overline{C_{7}}-\overline{C_{8}}+\overline{C_{9}}-\overline{C_{10}}]\),
\(e_{4}=\frac{1}{48}[1-\overline{C_{2}}+\overline{C_{3}}+\overline{C_{4}}-\overline{C_{5}}+\overline{C_{6}}-\overline{C_{7}}+\overline{C_{8}}-\overline{C_{9}}+\overline{C_{10}}]\)
where \(\overline{C_{i}}=\sum\limits_{g\in C_{i}}^{} g\).
Lemma 4.2. The nonlinear idempotents in \(\mathscr{F}[\overrightarrow{S_{3}}]\) are given by
\(v_{1}=\frac{1}{48}[(2)1-\overline{C_{3}}+(2)\overline{C_{4}}+(2)\overline{C_{6}}+(2)\overline{C_{8}}-\overline{C_{10}}],\)
\(v_{2}=\frac{1}{48}[(2)1-\overline{C_{3}}-(2)\overline{C_{4}}+(2)\overline{C_{6}}-(2)\overline{C_{8}}+\overline{C_{10}}],\)
\(v_{3}=\frac{1}{48}[(3)1+\overline{C_{2}}+\overline{C_{4}}-\overline{C_{5}}-\overline{C_{6}}+\overline{C_{7}}-(3)\overline{C_{8}}-\overline{C_{9}}],\)
\(v_{4}=\frac{1}{48}[(3)1-\overline{C_{2}}-\overline{C_{4}}-\overline{C_{5}}-\overline{C_{6}}+\overline{C_{7}}+(3)\overline{C_{8}}+\overline{C_{9}}],\)
\(v_{5}=\frac{1}{48}[(3)1+\overline{C_{2}}-\overline{C_{4}}+\overline{C_{5}}-\overline{C_{6}}-\overline{C_{7}}+(3)\overline{C_{8}}-\overline{C_{9}}],\)
\(v_{6}=\frac{1}{48}[(3)1-\overline{C_{2}}+\overline{C_{4}}+\overline{C_{5}}-\overline{C_{6}}-\overline{C_{7}}-(3)\overline{C_{8}}+\overline{C_{9}}]\)
where \(\overline{C_{i}}=\sum\limits_{g\in C_{i}}^{} g\).
Further, we compute the product of an arbitrary element and the idempotents in \(\mathscr{F}[\overrightarrow{S_{3}}]\) which will be used to construct various group codes in \(\mathscr{F}[\overrightarrow{S_{3}}]\). For this, let \(u=\sum\limits_{i=1}^{48} \lambda_{i}g_{i}\) be an arbitrary element in \(\mathscr{F}[\overrightarrow{S_{3}}]\) with \(\lambda_{i}\in \mathscr{F}\) for \(i=1, 2, …, 48\), then calculating by SageMath [22], we have
\[\ ue_{1}=\Big(\sum\limits_{i=1}^{48} \lambda_{i}\Big)e_{1},\label{p12}\tag{5}\] \[ue_{2}=\Big(\sum\limits_{i\in \mathcal{S}_{2}}\lambda_{i}-\sum\limits_{i\notin \mathcal{S}_{2}}\lambda_{i}\Big)e_{2},\label{p22}\tag{6}\] \[ue_{3}=\Big(\sum\limits_{i\in \mathcal{S}_{3}}\lambda_{i}-\sum\limits_{i\notin \mathcal{S}_{3}}\lambda_{i}\Big)e_{3},\label{p32}\tag{7}\] \[ue_{4}=\Big(\sum\limits_{i\in \mathcal{S}_{4}}\lambda_{i}-\sum\limits_{i\notin \mathcal{S}_{4}}\lambda_{i}\Big)e_{4},\label{p42} \tag{8}\] where \(\mathcal{S}_{2}=\{1,8,9,…,15,19,20,…,33\},\) \(\mathcal{S}_{3}=\{1,2,…,15,25,26,27,35,36,…,40\}\) and\(\mathcal{S}_{4}=\{1,8,9,…,18, 25,26,27,34,41,42,…,48\}\).
Furthermore, Using SageMath [22] the products corresponding to the nonlinear idempotents are
\(uv_{1}=\frac{1}{48}\big[ 2 \lambda_{ 1
}-\lambda_{ 8 }-\lambda_{ 9 }-\lambda_{ 10 }-\lambda_{ 11 }-\lambda_{ 12
}-\lambda_{ 13 }-\lambda_{ 14 }-\lambda_{ 15 }+ 2 \lambda_{ 16 }+ 2
\lambda_{ 17 }+ 2 \lambda_{ 18 }+ 2 \lambda_{ 25 }+ 2 \lambda_{ 26 }+ 2
\lambda_{ 27 }+ 2 \lambda_{ 34 }\)\(-\lambda_{ 41 }-\lambda_{ 42 }-\lambda_{
43 }-\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{ 47
}-\lambda_{ 48 }\big]\big(g_{ 1 }+g_{ 16 }+g_{ 17 }+g_{ 18 }+g_{ 25
}+g_{ 26 }+g_{ 27 }+g_{ 34 }\big)+%1
\big[ 2 \lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }\)\(-\lambda_{ 5
}-\lambda_{ 6 }+ 2 \lambda_{ 7 }+ 2 \lambda_{ 19 }-\lambda_{ 20
}-\lambda_{ 21 }-\lambda_{ 22 }-\lambda_{ 23 }+ 2 \lambda_{ 24
}-\lambda_{ 28 }-\lambda_{ 29 }-\lambda_{ 30 }+ 2 \lambda_{ 31
}-\lambda_{ 32 }+ 2 \lambda_{ 33 }+ 2 \lambda_{ 35 }-\lambda_{ 36 }+ 2
\lambda_{ 37 }\)\(-\lambda_{ 38 }-\lambda_{ 39 }-\lambda_{ 40 }\big]\big(g_{
2 }+g_{ 7 }+g_{ 19 }+g_{ 24 }+g_{ 31 }+g_{ 33 }+g_{ 35 }+g_{ 37
}\big)+%2
\big[-\lambda_{ 2 }+ 2 \lambda_{ 3 }-\lambda_{ 4 }+ 2 \lambda_{ 5
}-\lambda_{ 6 }-\lambda_{ 7 }-\lambda_{ 19 }-\lambda_{ 20 }\)\(+ 2 \lambda_{
21 }-\lambda_{ 22 }+ 2 \lambda_{ 23 }-\lambda_{ 24 }+ 2 \lambda_{ 28 }+
2 \lambda_{ 29 }-\lambda_{ 30 }-\lambda_{ 31 }-\lambda_{ 32 }-\lambda_{
33 }-\lambda_{ 35 }-\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }+ 2
\lambda_{ 39 }+ 2 \lambda_{ 40 }\big]\big(g_{ 3 }+g_{ 5 }\)\(+g_{ 21 }+g_{
23 }+g_{ 28 }+g_{ 29 }+g_{ 39 }+g_{ 40 }\big)+%3
\big[-\lambda_{ 2 }-\lambda_{ 3 }+ 2 \lambda_{ 4 }-\lambda_{ 5 }+ 2
\lambda_{ 6 }-\lambda_{ 7 }-\lambda_{ 19 }+ 2 \lambda_{ 20 }-\lambda_{
21 }+ 2 \lambda_{ 22 }-\lambda_{ 23 }-\lambda_{ 24 }\)\(-\lambda_{ 28
}-\lambda_{ 29 }+ 2 \lambda_{ 30 }-\lambda_{ 31 }+ 2 \lambda_{ 32
}-\lambda_{ 33 }-\lambda_{ 35 }+ 2 \lambda_{ 36 }-\lambda_{ 37 }+ 2
\lambda_{ 38 }-\lambda_{ 39 }-\lambda_{ 40 }\big]\big(g_{ 4 }+g_{ 6
}+g_{ 20 }+g_{ 22 }+g_{ 30 }+g_{ 32 }\)\(+g_{ 36 }+g_{ 38 }\big)+%4
\big[-\lambda_{ 1 }+ 2 \lambda_{ 8 }-\lambda_{ 9 }+ 2 \lambda_{ 10
}-\lambda_{ 11 }-\lambda_{ 12 }-\lambda_{ 13 }+ 2 \lambda_{ 14 }+ 2
\lambda_{ 15 }-\lambda_{ 16 }-\lambda_{ 17 }-\lambda_{ 18 }-\lambda_{ 25
}-\lambda_{ 26 }-\lambda_{ 27 }\)\(-\lambda_{ 34 }+ 2 \lambda_{ 41 }+ 2
\lambda_{ 42 }-\lambda_{ 43 }-\lambda_{ 44 }-\lambda_{ 45 }+ 2 \lambda_{
46 }-\lambda_{ 47 }+ 2 \lambda_{ 48 }\big]\big(g_{ 8 }+g_{ 10 }+g_{ 14
}+g_{ 15 }+g_{ 41 }+g_{ 42 }+g_{ 46 }+g_{ 48 }\big)+%5
\)\(\big[-\lambda_{ 1 }-\lambda_{ 8 }+ 2 \lambda_{ 9 }-\lambda_{ 10 }+ 2
\lambda_{ 11 }+ 2 \lambda_{ 12 }+ 2 \lambda_{ 13 }-\lambda_{ 14
}-\lambda_{ 15 }-\lambda_{ 16 }-\lambda_{ 17 }-\lambda_{ 18 }-\lambda_{
25 }-\lambda_{ 26 }-\lambda_{ 27 }-\lambda_{ 34 }-\lambda_{ 41
}-\lambda_{ 42 }\)\(+ 2 \lambda_{ 43 }+ 2 \lambda_{ 44 }+ 2 \lambda_{ 45
}-\lambda_{ 46 }+ 2 \lambda_{ 47 }-\lambda_{ 48 }\big]\big(g_{ 9 }+g_{
11 }+g_{ 12 }+g_{ 13 }+g_{ 43 }+g_{ 44 }+g_{ 45 }+g_{ 47
}\big)\).
Similarly, the products \(uv_{2}, uv_{3},
uv_{4}, uv_{5}\) and \(uv_{6}\)
have been obtained in Appendix I.
Next, in support of Theorem 2.4, we present the following example for \(n=3\).
Example 4.3. Let \(e_{1}\), \(e_{2}\), \(e_{3}\), \(e_{4}\) be linear idempotents in \(\mathscr{F}[\overrightarrow{S_{3}}]\). Then \(\mathscr{I}_{\{e_{i}\}}\) for \(i=1,2,3,4\), is \((48, 47, 2)\) MDS group code.
The result stated below can also be derived from Theorem 2.5. However, we are giving the proof of this result for better understanding, so that we can use equations(9) to (12) in the later part of this paper.
Theorem 4.4. Let \(e_{1}\), \(e_{2}\), \(e_{3}\), \(e_{4}\) be linear idempotents in \(\mathscr{F}[\overrightarrow{S_{3}}]\) and \(i, j, k\) be three distinct elements of the set \(\{1, 2, 3, 4\}\). Then
\((i)\) \(\mathscr{I}_{\{e_{i}, e_{j}\}}\) is \((48, 46, 2)\) group code.
\((ii)\) \(\mathscr{I}_{\{e_{i}, e_{j}, e_{k}\}}\) is \((48, 45, 2)\) group code.
\((iii)\) \(\mathscr{I}_{\{e_{1}, e_{2}, e_{3}, e_{4}\}}\) is \((48, 44, 2)\) group code.
Proof. Assume that \(\mathscr{I}\) denotes the corresponding ideal in \((i)\), \((ii)\) and \((iii)\). First, observe that it follows from (5) to (8) that \(ue_{i}=0\) for \(i\in\{1,2,3,4\}\) if and only if the \(i^{th}\) equation in the following equations is satisfied. \[\begin{gathered} \label{fp12=0} \ \lambda_{ 1 }+\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{ 5 }+\lambda_{ 6 }+\lambda_{ 7 }+\lambda_{ 8 }+\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }+\lambda_{ 12 }+\lambda_{ 13 }+\lambda_{ 14 }+\lambda_{ 15 }+\lambda_{ 16 }+\lambda_{ 17 }+\lambda_{ 18 }\\ +\lambda_{ 19 }+\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{ 22 }+\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 25 }+\lambda_{ 26 }+\lambda_{ 27 }+ \lambda_{ 28 }+\lambda_{ 29 }+\lambda_{ 30 }+\lambda_{ 31 }+\lambda_{ 32 }+\lambda_{ 33 }\\ +\lambda_{ 34 }+\lambda_{ 35 }+\lambda_{ 36 }+\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39 }+\lambda_{ 40 }+ \lambda_{ 41 }+\lambda_{ 42 }+\lambda_{ 43 }+\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{ 47 }+\lambda_{ 48 }=0, \end{gathered} \tag{9}\] \[\begin{gathered} \label{fp22=0} \lambda_{ 1 }-\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{ 5 }-\lambda_{ 6 }-\lambda_{ 7 }+\lambda_{ 8 }+\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }+\lambda_{ 12 }+\lambda_{ 13 }+\lambda_{ 14 }+\lambda_{ 15 }-\lambda_{ 16 }-\\ \lambda_{ 17 }-\lambda_{ 18 }+\lambda_{ 19 }+\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{ 22 }+\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 25 }+\lambda_{ 26 }+\lambda_{ 27 }+ \lambda_{ 28 }+\lambda_{ 29 }+\lambda_{ 30 }+\lambda_{ 31 }+\lambda_{ 32 }+\\ \lambda_{ 33 }-\lambda_{ 34 }-\lambda_{ 35 }-\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39 }-\lambda_{ 40 }- \lambda_{ 41 }-\lambda_{ 42 }-\lambda_{ 43 }-\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{ 47 }-\lambda_{ 48 }=0, \end{gathered} \tag{10}\] \[\begin{gathered} \label{fp32=0} \lambda_{ 1 }+\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{ 5 }+\lambda_{ 6 }+\lambda_{ 7 }+\lambda_{ 8 }+\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }+\lambda_{ 12 }+\lambda_{ 13 }+\lambda_{ 14 }+ \lambda_{ 15 }-\lambda_{ 16 }-\\ \lambda_{ 17 }-\lambda_{ 18 }-\lambda_{ 19 }-\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{ 22 }-\lambda_{ 23 }-\lambda_{ 24 }+\lambda_{ 25 }+\lambda_{ 26 }+\lambda_{ 27 }- \lambda_{ 28 }-\lambda_{ 29 }-\lambda_{ 30 }-\lambda_{ 31 }-\lambda_{ 32 }-\\ \lambda_{ 33 }-\lambda_{ 34 }+\lambda_{ 35 }+\lambda_{ 36 }+\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39 }+\lambda_{ 40 }- \lambda_{ 41 }-\lambda_{ 42 }-\lambda_{ 43 }-\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{ 47 }-\lambda_{ 48 }=0, \end{gathered} \tag{11}\] \[\begin{gathered} \label{fp42=0} \lambda_{ 1 }-\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{ 5 }-\lambda_{ 6 }-\lambda_{ 7 }+\lambda_{ 8 }+\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }+\lambda_{ 12 }+\lambda_{ 13 }+\lambda_{ 14 }+ \lambda_{ 15 }+\lambda_{ 16 }+\\ \lambda_{ 17 }+\lambda_{ 18 }-\lambda_{ 19 }-\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{ 22 }-\lambda_{ 23 }-\lambda_{ 24 }+\lambda_{ 25 }+\lambda_{ 26 }+\lambda_{ 27 }- \lambda_{ 28 }-\lambda_{ 29 }-\lambda_{ 30 }-\lambda_{ 31 }-\lambda_{ 32 }-\\ \lambda_{ 33 }+\lambda_{ 34 }-\lambda_{ 35 }-\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39 }-\lambda_{ 40 }+ \lambda_{ 41 }+\lambda_{ 42 }+\lambda_{ 43 }+\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{ 47 }+\lambda_{ 48 }=0. \end{gathered} \tag{12}\] Now, in any of the parts \((i)\), \((ii)\) and \((iii)\), if we choose an element \(u=\lambda g\), for \(g\in\overrightarrow{S_{3}}\), \(0\neq\lambda\in \mathscr{F}\) with \(wt(u)=1\), then it follows from equations (9) to (12) that \(ue_{i}=0\) for \(i=1,2,3,4,\) if and only if \(\lambda=0\). This implies that \(u\notin \mathscr{I}\) and so \(d(\mathscr{I})\geq 2\). We claim that for any two distinct choices of elements \(s\), \(t\) from the set \(\{19, 20, …, 24, 28, 29, …, 33\}\), there exists an element \(u=g_{s}-g_{t}\in \mathscr{F}[\overrightarrow{S_{3}}]\) such that \(u\in \mathscr{I}\). For this, it is enough to show that \(ue_{i}=0\), for each \(i=1, 2, 3, 4\). Clearly, for \(u=g_{s}-g_{t}\), we have \(\lambda_{g_{s}}=1, \lambda_{g_{t}}=-1\) and \(\lambda_{g_{k}}=0\), for all \(k\in\{1,2,…,48\}\setminus\{s,t\}\). Therefore, by using equation (9) to (12), we have \(ue_{i}=(1-1)e_{i}=0\), for \(i=1, 2, 3,4\). Thus, \(d(\mathscr{I})=2\). Moreover, the dimension of ideals \(\mathscr{I}\) over \(\mathscr{F}\) in the case \((i)\), \((ii)\) or \((iii)\) is \(46\), \(45\) or \(44\) respectively. Hence the result. ◻
The upcoming theorems provide us with various group codes in \(\mathscr{F}[\overrightarrow{S_{3}}]\), which are generated nonlinear idempotents or a combination of linear and nonlinear idempotents in \(\mathscr{F}[\overrightarrow{S_{3}}]\).
Let \(v_{1}\) and \(v_{2}\) be nonlinear idempotents in \(\mathscr{F}[\overrightarrow{S_{3}}]\) corresponding to nonlinear characters of degree \(2\). Then
\((i)\) \(\mathscr{I}_{\{v_{i}\}}\) for \(i=1, 2\), is \((48, 44, 2)\) group code.
\((ii)\) \(\mathscr{I}_{\{v_{1}, v_{2}\}}\) is \((48, 40, 2)\) group code.
Proof. Let \(u=\sum\limits_{i=1}^{48} \lambda_{i}g_{i}\in \mathscr{F}[\overrightarrow{S_{3}}]\) and \(\mu\subseteq\{v_{1}, v_{2}\}\). Then \(\mathscr{I}_{\mu}=\{u\in \mathscr{F}[\overrightarrow{S_{3}}] : ux=0\ \text{for all}\ x\in\mu\}\). Note that \(uv_{1}=0\) if and only if the following equations are satisfied.
\[\begin{aligned} \label{fp52:1} &\lambda_{ 1 }+\lambda_{ 8 }-2 \lambda_{ 9 }+\lambda_{ 10 }-2 \lambda_{ 11 }-2 \lambda_{ 12 }-2 \lambda_{ 13 }+\lambda_{ 14 }+\lambda_{ 15 }+\lambda_{ 16 }+\lambda_{ 17 }+\lambda_{ 18 }+\lambda_{ 25 } \notag \\ &+\lambda_{ 26 }+\lambda_{ 27 }+\lambda_{ 34 }+\lambda_{ 41 }+\lambda_{ 42 }-2 \lambda_{ 43 }-2 \lambda_{ 44 }-2 \lambda_{ 45 }+\lambda_{ 46 }-2 \lambda_{ 47 }+\lambda_{ 48 }=0, \end{aligned} \tag{13}\]
\[\begin{aligned} \label{fp52:2} &\lambda_{ 2 }+\lambda_{ 3 }-2 \lambda_{ 4 }+\lambda_{ 5 }-2 \lambda_{ 6 }+\lambda_{ 7 }+\lambda_{ 19 }-2 \lambda_{ 20 }+\lambda_{ 21 }-2 \lambda_{ 22 }+\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 28 }\notag\\ &+ \lambda_{ 29 }-2 \lambda_{ 30 }+\lambda_{ 31 }-2 \lambda_{ 32 }+\lambda_{ 33 }+\lambda_{ 35 }-2 \lambda_{ 36 }+\lambda_{ 37 }-2 \lambda_{ 38 }+\lambda_{ 39 }+\lambda_{ 40 }=0, \end{aligned} \tag{14}\]
\[\begin{aligned} \label{fp52:3} &3 \lambda_{ 3 }-3 \lambda_{ 4 }+ 3 \lambda_{ 5 }-3 \lambda_{ 6 }-3 \lambda_{ 20 }+ 3 \lambda_{ 21 }-3 \lambda_{ 22 }+ 3 \lambda_{ 23 }+ 3 \lambda_{ 28 }+ 3 \lambda_{ 29 }-3 \lambda_{ 30 }\notag\\ &- 3 \lambda_{ 32 }-3 \lambda_{ 36 }-3 \lambda_{ 38 }+ 3 \lambda_{ 39 }+ 3 \lambda_{ 40 }=0, \end{aligned} \tag{15}\]
\[\begin{aligned} \label{fp52:4} &3 \lambda_{ 8 }-3 \lambda_{ 9 }+ 3 \lambda_{ 10 }-3 \lambda_{ 11 }-3 \lambda_{ 12 }-3 \lambda_{ 13 }+ 3 \lambda_{ 14 }+ 3 \lambda_{ 15 }+ 3 \lambda_{ 41 }+ 3 \lambda_{ 42 }-3 \lambda_{ 43 }\notag\\ &- 3 \lambda_{ 44 }-3 \lambda_{ 45 }+ 3 \lambda_{ 46 }-3 \lambda_{ 47 }+ 3 \lambda_{ 48 }=0, \end{aligned} \tag{16}\] whereas, \(uv_{2}=0\) if and only if the following equations are satisfied.
\[\begin{aligned} \label{fp62:1} &\lambda_{ 1 }+\lambda_{ 8 }-2 \lambda_{ 9 }+\lambda_{ 10 }-2 \lambda_{ 11 }-2 \lambda_{ 12 }-2 \lambda_{ 13 }+\lambda_{ 14 }+\lambda_{ 15 }-\lambda_{ 16 }-\lambda_{ 17 }-\lambda_{ 18 }+\lambda_{ 25 }\notag\\ & +\lambda_{ 26 }+\lambda_{ 27 }-\lambda_{ 34 }-\lambda_{ 41 }-\lambda_{ 42 }+ 2 \lambda_{ 43 }+ 2 \lambda_{ 44 }+ 2 \lambda_{ 45 }-\lambda_{ 46 }+ 2 \lambda_{ 47 }-\lambda_{ 48 }=0, \end{aligned} \tag{17}\]
\[\begin{aligned} \label{fp62:2} &\lambda_{ 2 }+\lambda_{ 3 }-2 \lambda_{ 4 }+\lambda_{ 5 }-2 \lambda_{ 6 }+\lambda_{ 7 }-\lambda_{ 19 }+ 2 \lambda_{ 20 }-\lambda_{ 21 }+ 2 \lambda_{ 22 }-\lambda_{ 23 }-\lambda_{ 24 }-\lambda_{ 28 }\notag \\ &- \lambda_{ 29 }+ 2 \lambda_{ 30 }-\lambda_{ 31 }+ 2 \lambda_{ 32 }-\lambda_{ 33 }+\lambda_{ 35 }-2 \lambda_{ 36 }+\lambda_{ 37 }-2 \lambda_{ 38 }+\lambda_{ 39 }+\lambda_{ 40 }=0 , \end{aligned} \tag{18}\]
\[\begin{aligned} \label{fp62:3} &3 \lambda_{ 3 }-3 \lambda_{ 4 }+ 3 \lambda_{ 5 }-3 \lambda_{ 6 }+ 3 \lambda_{ 20 }-3 \lambda_{ 21 }+ 3 \lambda_{ 22 }-3 \lambda_{ 23 }-3 \lambda_{ 28 }-3 \lambda_{ 29 }+ 3 \lambda_{ 30 }\notag \\ &+ 3 \lambda_{ 32 }-3 \lambda_{ 36 }-3 \lambda_{ 38 }+ 3 \lambda_{ 39 }+ 3 \lambda_{ 40 }=0 , \end{aligned} \tag{19}\]
\[\begin{aligned} \label{fp62:4} &3 \lambda_{ 8 }-3 \lambda_{ 9 }+ 3 \lambda_{ 10 }-3 \lambda_{ 11 }-3 \lambda_{ 12 }-3 \lambda_{ 13 }+ 3 \lambda_{ 14 }+ 3 \lambda_{ 15 }-3 \lambda_{ 41 }-3 \lambda_{ 42 }+ 3 \lambda_{ 43 }\notag \\ &+ 3 \lambda_{ 44 }+ 3 \lambda_{ 45 }-3 \lambda_{ 46 }+ 3 \lambda_{ 47 }-3 \lambda_{ 48 }=0 . \end{aligned} \tag{20}\]
\((i)\) Let \(I_{\{v_{i}\}}\) be the set of elements of the form \(u=\sum\limits_{i=1}^{48} \lambda_{i}g_{i}\in \mathscr{F}[\overrightarrow{S_{3}}]\) such that the coefficients in the presentation of \(u\) satisfy (13) to (16), if \(i=1\), while equation (17) to (20), if \(i=2\). Now, as observed in the previous theorem, we can similarly prove that \(\mathscr{I}_{\{v_{i}\}}\) does not contain any element with weight \(1\). Also, an element with the minimum weight in \(\mathscr{I}_{\{v_{i}\}}\), for \(i=1,2\), is \(u=g_{3}-g_{5}\) with weight \(2\), as the coefficients of \(\lambda_{3}\) and \(\lambda_{5}\) in equation (13) to (20) are equal. Thus, we conclude that \(d(\mathscr{I}_{\{v_{i}\}})=2\). Moreover, the dimension of \(\mathscr{I}_{\{v_{i}\}}\) is \(44\). Hence \(\mathscr{I}_{\{v_{i}\}}\) is \((48, 44, 2)\) group code.
\((ii)\) Consider an ideal \(\mathscr{I}_{\{v_{1}, v_{2}\}}\) of \(\mathscr{F}[\overrightarrow{S_{3}}]\) which consists of the elements of the form \(u=\sum\limits_{i=1}^{48} \lambda_{i}g_{i}\in \mathscr{F}[\overrightarrow{S_{3}}]\) such that the coefficients in the presentation of \(u\) satisfy equation (13) to (20). Again, one can realize that \(\mathscr{I}_{\{v_{1}, v_{2}\}}\) does not contain any element with weight \(1\) and an element with the minimum weight in \(\mathscr{I}_{\{v_{1}, v_{2}\}}\) is \(u=g_{3}-g_{5}\) with its weight \(2\). Therefore, \(d(\mathscr{I}_{\{v_{1}, v_{2}\}})=2\). Moreover, the dimension of \(\mathscr{I}_{\{v_{1}, v_{2}\}}\) is \(40\). Hence \(\mathscr{I}_{\{v_{1}, v_{2}\}}\) is \((48, 40, 2)\) group code. ◻
Theorem 4.6. Let \(e_{1}\), \(e_{2}\), \(e_{3}\), \(e_{4}\) be linear idempotents and \(v_{1}\), \(v_{2}\) be nonlinear idempotents corresponding to nonlinear characters of degree \(2\) of \(\mathscr{F}[\overrightarrow{S_{3}}]\). If \(X_{i}\) and \(Y_{j}\) are subsets of order \(i\) and \(j\) of the sets \(\{e_{1}, e_{2}, e_{3}, e_{4}\}\) and \(\{v_{1}, v_{2}\}\) respectively. Then
\((i)\) \(\mathscr{I}_{X_{1}\cup Y_{1}}\) is \((48, 43, 2)\) group code.
\((ii)\) \(\mathscr{I}_{X_{1}\cup Y_{2}}\) is \((48, 39, 2)\) group code.
\((iii)\) \(\mathscr{I}_{X_{2}\cup Y_{1}}\) is \((48, 42, 2)\) group code.
\((iv)\) \(\mathscr{I}_{X_{2}\cup Y_{2}}\) is \((48, 38, 2)\) group code.
\((v)\) \(\mathscr{I}_{X_{3}\cup Y_{1}}\) is \((48, 41, 2)\) group code.
\((vi)\) \(\mathscr{I}_{X_{3}\cup Y_{2}}\) is \((48, 37, 2)\) group code.
\((vii)\) \(\mathscr{I}_{X_{4}\cup Y_{1}}\) is \((48, 40, 2)\) group code.
\((viii)\) \(\mathscr{I}_{X_{4}\cup Y_{2}}\) is \((48, 36, 2)\) group code.
Proof. Assume that \(\mathscr{I}_{X_{i}\cup Y_{j}}\) denotes the corresponding ideal in any possible case of \((i)\), \((ii)\), \((iii)\) and \((iv)\), where \(X_{i}\) and \(Y_{j}\) are some subsets of order \(i\) and \(j\) of the sets \(\{e_{1}, e_{2}, e_{3}, e_{4}\}\) and \(\{v_{1}, v_{2}\}\) respectively. We claim that \(d(\mathscr{I})=2\). For this, consider an ideal \(\mathscr{I}_{X_{i}\cup Y_{j}}\) of \(\mathscr{F}[\overrightarrow{S_{3}}]\) containing the elements of the form \(u=\sum\limits_{i=1}^{48} \lambda_{i}g_{i}\) such that \(ux=0\), for all \(x\in X_{i}\cup Y_{j}\). This infers that depending upon the subsets \(X_{i}\) and \(Y_{j}\) the coefficients in the presentation of \(u\) must satisfy suitable equations from equation (9) to (20), corresponding to \(ux=0\), for all \(x\in X_{i}\cup Y_{j}\). Clearly, \(\mathscr{I}_{X_{i}\cup Y_{j}}\) can not contain any element of weight \(1\) and so \(d(\mathscr{I}_{X_{i}\cup Y_{j}})\geq 2\). Further, an element with the smallest weight \(2\) in \(\mathscr{I}_{X_{i}\cup Y_{j}}\) is \(u=g_{3}-g_{5}\), as the coefficient of \(\lambda_{3}\) and \(\lambda_{5}\) in all the equation (9) to (20) are equal. Thus, we conclude that \(d(\mathscr{I}_{X_{i}\cup Y_{j}})=2\). Next, for the dimension of \(\mathscr{I}_{X_{i}\cup Y_{j}}\), observe that all the equation (9) to (20) are linearly independent therefore, the dimension of \(\mathscr{I}_{X_{i}\cup Y_{j}}\) is \(48-(i+4j)\). This completes the proof. ◻
Theorem 4.7. Let \(v_{3}\), \(v_{4}\), \(v_{5}\) and \(v_{6}\) be nonlinear idempotents in \(\mathscr{F}[\overrightarrow{S_{3}}]\) corresponding to nonlinear characters of degree \(3\). Then
\((i)\) \(\mathscr{I}_{\{v_{i}\}}\) for \(i=3,4,5,6\), is \((48, 39, 2)\) group code.
\((ii)\) \(\mathscr{I}_{\{v_{3}, v_{6}\}}\) is \((48, 30, 2)\) group code.
\((iii)\) \(\mathscr{I}_{\{v_{4}, v_{5}\}}\) is \((48, 30, 2)\) group code.
Proof. Let \(u=\sum\limits_{i=1}^{48} \lambda_{i}g_{i}\in \mathscr{F}[\overrightarrow{S_{3}}]\) and \(\mu\subseteq\{v_{3}, v_{4}, v_{5}, v_{6}\}\). Then \(\mathscr{I}_{\mu}=\{u\in \mathscr{F}[\overrightarrow{S_{3}}] : ux=0\ \text{for all}\ x\in\mu\}\). Note that \(uv_{3}=0\) if and only if the following equations are satisfied.
\[\begin{aligned} \label{fp662:1} &\lambda_{ 1 }+ 2 \lambda_{ 5 }+ 2 \lambda_{ 6 }-2 \lambda_{ 7 }-\lambda_{ 9 }+ 3 \lambda_{ 11 }+ 3 \lambda_{ 12 }+ 4 \lambda_{ 14 }+ 3 \lambda_{ 16 }-\lambda_{ 17 }-\lambda_{ 18 }+2 \lambda_{ 19 }- 2 \lambda_{ 20 }-2 \lambda_{ 21 }+\lambda_{ 25 }+\lambda_{ 26 }\notag \\ &-3 \lambda_{ 27 }+ 6 \lambda_{ 28 }+ 4 \lambda_{ 30 }-2 \lambda_{ 31 }-2 \lambda_{ 32 }- \lambda_{ 34 }+2 \lambda_{ 36 }+2 \lambda_{ 37 }+ 4 \lambda_{ 39 }+ 4 \lambda_{ 41 }+ 5 \lambda_{ 43 }-3 \lambda_{ 44 }-3 \lambda_{ 45 }+\lambda_{ 47 }=0 , \end{aligned} \tag{21}\]
\[\begin{aligned} \label{fp662:2} &\lambda_{ 2 }+ 2 \lambda_{ 5 }+ 2 \lambda_{ 6 }+\lambda_{ 9 }+ 5 \lambda_{ 11 }+ 3 \lambda_{ 12 }+ 6 \lambda_{ 14 }+ 4 \lambda_{ 16 }-2 \lambda_{ 17 }-2 \lambda_{ 18 }+5 \lambda_{ 19 }-2 \lambda_{ 20 } -2 \lambda_{ 21 }-\lambda_{ 24 }\notag\\ &+ 2 \lambda_{ 25 }+ 2 \lambda_{ 26 }+ 8 \lambda_{ 28 }+ 6 \lambda_{ 30 }+\lambda_{ 33 }-\lambda_{ 35 }+ 4 \lambda_{ 36 } + 5 \lambda_{ 37 }+ 6 \lambda_{ 39 }+6 \lambda_{ 41 }+ 9 \lambda_{ 43 }-3 \lambda_{ 44 }-\lambda_{ 47 }=0 , \end{aligned} \tag{22}\]
\[\begin{aligned} \label{fp662:3} &\lambda_{ 3 }+ 3 \lambda_{ 5 }+ 2 \lambda_{ 6 }+\lambda_{ 9 }+ 2 \lambda_{ 10 }+ 3 \lambda_{ 11 }+ 3 \lambda_{ 12 }+ 4 \lambda_{ 14 }+ 6 \lambda_{ 16 }-2 \lambda_{ 17 }+6 \lambda_{ 19 }-2 \lambda_{ 20 }-3 \lambda_{ 21 }-\lambda_{ 23 }+ 4 \lambda_{ 25 }\notag\\ &+ 2 \lambda_{ 26 }+ 7 \lambda_{ 28 }-3 \lambda_{ 29 }+ 6 \lambda_{ 30 }-2 \lambda_{ 33 }+ 2 \lambda_{ 35 }+4 \lambda_{ 36 }+ 4 \lambda_{ 37 }+ 3 \lambda_{ 39 }+ 6 \lambda_{ 41 }+ 7 \lambda_{ 43 }-3 \lambda_{ 44 }\notag\\ &-3 \lambda_{ 45 }-2 \lambda_{ 46 }-\lambda_{ 47 }=0 , \end{aligned} \tag{23}\]
\[\begin{aligned} \label{fp662:4} &\lambda_{ 4 }+ 2 \lambda_{ 5 }+ 3 \lambda_{ 6 }+\lambda_{ 9 }+ 2 \lambda_{ 10 }+ 3 \lambda_{ 11 }+\lambda_{ 12 }+ 2 \lambda_{ 14 }+ 6 \lambda_{ 16 }-2 \lambda_{ 18 }+ 6 \lambda_{ 19 }\notag\\& -3 \lambda_{ 20 }-2 \lambda_{ 21 }-\lambda_{ 22 }+ 2 \lambda_{ 25 }+ 4 \lambda_{ 26 }+ 4 \lambda_{ 28 }-2 \lambda_{ 29 }+ 5 \lambda_{ 30 }-\lambda_{ 32 }-2 \lambda_{ 33 }\notag\\& + 2 \lambda_{ 35 }+ \lambda_{ 36 } +4 \lambda_{ 37 }+ 2 \lambda_{ 39 }+ 4 \lambda_{ 41 }-2 \lambda_{ 42 }+ 5 \lambda_{ 43 }-\lambda_{ 44 }-3 \lambda_{ 45 }-2 \lambda_{ 46 }-\lambda_{ 47 }=0 , \end{aligned} \tag{24}\]
\[\begin{aligned} \label{fp662:5} 4 \lambda_{ 5 }+ 4 \lambda_{ 11 }+ 4 \lambda_{ 16 }+ 4 \lambda_{ 19 }+ 4 \lambda_{ 25 }+ 4 \lambda_{ 28 }+ 4 \lambda_{ 37 }+ 4 \lambda_{ 43 }=0 , \end{aligned} \tag{25}\]
\[\begin{aligned} \label{fp662:6} 4 \lambda_{ 6 }+ 4 \lambda_{ 11 }+ 4 \lambda_{ 14 }+ 4 \lambda_{ 16 }+ 4 \lambda_{ 19 }+ 4 \lambda_{ 26 }+ 4 \lambda_{ 28 }+ 4 \lambda_{ 30 }+ 4 \lambda_{ 37 }+ 4 \lambda_{ 39 }+ 4 \lambda_{ 41 }+ 4 \lambda_{ 43 }=0 , \end{aligned} \tag{26}\]
\[\begin{aligned} \label{fp662:7} &\lambda_{ 8 }-\lambda_{ 9 }+\lambda_{ 10 }-\lambda_{ 11 }+ 3 \lambda_{ 12 }-\lambda_{ 13 }+\lambda_{ 14 }-3 \lambda_{ 15 }+ 2 \lambda_{ 28 }-2 \lambda_{ 29 }+ 2 \lambda_{ 30 }+ 2 \lambda_{ 31 }-2 \lambda_{ 32 }-2 \lambda_{ 33 }\notag\\&+ 2 \lambda_{ 35 } + 2 \lambda_{ 36 }-2 \lambda_{ 37 }-2 \lambda_{ 38 }+ 2 \lambda_{ 39 }-2 \lambda_{ 40 }+3 \lambda_{ 41 } -\lambda_{ 42 }+\lambda_{ 43 }-3 \lambda_{ 44 }+\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{ 47 }-\lambda_{ 48 }=0 , \end{aligned} \tag{27}\]
\[\begin{aligned} \label{fp662:8} 4 \lambda_{ 10 }+ 4 \lambda_{ 13 }+ 4 \lambda_{ 29 }+ 4 \lambda_{ 31 }+ 4 \lambda_{ 35 }+ 4 \lambda_{ 40 }+ 4 \lambda_{ 42 }+ 4 \lambda_{ 45 }=0 , \end{aligned} \tag{28}\]
\[\begin{aligned} \label{fp662:9} 4 \lambda_{ 12 }+ 4 \lambda_{ 14 }+ 4 \lambda_{ 28 }+ 4 \lambda_{ 30 }+ 4 \lambda_{ 36 }+ 4 \lambda_{ 39 }+ 4 \lambda_{ 41 }+ 4 \lambda_{ 43 }=0 , \end{aligned} \tag{29}\] whereas, \(uv_{4}=0\) if and only if the following equations are satisfied.
\[\begin{aligned} \label{fp72:1} &\lambda_{ 1 }+ 2 \lambda_{ 5 }+ 2 \lambda_{ 6 }-\lambda_{ 9 }+ 3 \lambda_{ 12 }+ 3 \lambda_{ 13 }+ 5 \lambda_{ 16 }-3 \lambda_{ 17 }-3 \lambda_{ 18 }+ 2 \lambda_{ 20 }+ 2 \lambda_{ 21 }-3 \lambda_{ 25 }-3 \lambda_{ 26 }\notag\\ &+ 5 \lambda_{ 27 }-2 \lambda_{ 28 }+ 6 \lambda_{ 31 }-2 \lambda_{ 32 }+\lambda_{ 34 }-2 \lambda_{ 36 }+ 6 \lambda_{ 37 }-2 \lambda_{ 40 }+3 \lambda_{ 43 }+ 3 \lambda_{ 44 }-\lambda_{ 47 }=0 , \end{aligned} \tag{30}\]
\[\begin{aligned} \label{fp72:2} &\lambda_{ 2 }+ 2 \lambda_{ 5 }+ 2 \lambda_{ 6 }-\lambda_{ 9 }+\lambda_{ 12 }+ 5 \lambda_{ 13 }-2 \lambda_{ 14 }+ 2 \lambda_{ 15 }+ 4 \lambda_{ 16 }-2 \lambda_{ 17 }- 2 \lambda_{ 18 }+2 \lambda_{ 20 }+ 2 \lambda_{ 21 }\notag\\&+\lambda_{ 24 }-2 \lambda_{ 25 }-2 \lambda_{ 26 }+ 4 \lambda_{ 27 }+ 2 \lambda_{ 29 }-2 \lambda_{ 30 }+5 \lambda_{ 31 }-\lambda_{ 33 }-\lambda_{ 35 }+ 5 \lambda_{ 37 }-2 \lambda_{ 38 }+ 2 \lambda_{ 39 }\notag\\&+ 2 \lambda_{ 41 }-2 \lambda_{ 42 }+ 5 \lambda_{ 43 }+\lambda_{ 44 }-\lambda_{ 47 }=0 , \end{aligned} \tag{31}\]
\[\begin{aligned} \label{fp72:3} \lambda_{ 3 }+ 3 \lambda_{ 5 }+ 2 \lambda_{ 6 }-\lambda_{ 9 }+ 2 \lambda_{ 10 }+\lambda_{ 12 }+ 7 \lambda_{ 13 }+ 2 \lambda_{ 15 }+ 6 \lambda_{ 16 }-2 \lambda_{ 17 }+ 2 \lambda_{ 20 }+3 \lambda_{ 21 }+\lambda_{ 23 }-2 \lambda_{ 26 }+ 6 \lambda_{ 27 }\notag\\ + 3 \lambda_{ 29 }-2 \lambda_{ 30 }+ 8 \lambda_{ 31 }-2 \lambda_{ 33 }-2 \lambda_{ 35 }+ 8 \lambda_{ 37 }-2 \lambda_{ 38 }+ 3 \lambda_{ 39 }+ 2 \lambda_{ 41 }+ 7 \lambda_{ 43 }+\lambda_{ 44 }+ 2 \lambda_{ 46 }-\lambda_{ 47 }=0 , \end{aligned} \tag{32}\]
\[\begin{aligned} \label{fp72:4} &\lambda_{ 4 }+ 2 \lambda_{ 5 }+ 3 \lambda_{ 6 }-\lambda_{ 9 }+ 2 \lambda_{ 10 }+ 3 \lambda_{ 12 }+ 5 \lambda_{ 13 }-2 \lambda_{ 14 }+ 6 \lambda_{ 16 }-2 \lambda_{ 18 }+ 3 \lambda_{ 20 }+2 \lambda_{ 21 }+\lambda_{ 22 }\notag\\&-2 \lambda_{ 25 }+ 6 \lambda_{ 27 }+ 2 \lambda_{ 29 } -\lambda_{ 30 }+ 8 \lambda_{ 31 }-3 \lambda_{ 32 }-2 \lambda_{ 33 }-2 \lambda_{ 35 }-3 \lambda_{ 36 }+ 8 \lambda_{ 37 }-\lambda_{ 38 }+ 2 \lambda_{ 39 }\notag\\&-2 \lambda_{ 42 }+ 5 \lambda_{ 43 }+ 3 \lambda_{ 44 }+ 2 \lambda_{ 46 }-\lambda_{ 47 }=0 , \end{aligned} \tag{33}\]
\[\begin{aligned} \label{fp72:5} 4 \lambda_{ 5 }+ 4 \lambda_{ 13 }+ 4 \lambda_{ 16 }+ 4 \lambda_{ 21 }+ 4 \lambda_{ 27 }+ 4 \lambda_{ 31 }+ 4 \lambda_{ 37 }+ 4 \lambda_{ 43 }=0 , \end{aligned} \tag{34}\]
\[\begin{aligned} \label{fp72:6} 4 \lambda_{ 6 }+ 4 \lambda_{ 13 }+ 4 \lambda_{ 15 }+ 4 \lambda_{ 16 }+ 4 \lambda_{ 20 }+ 4 \lambda_{ 27 }+ 4 \lambda_{ 29 }+ 4 \lambda_{ 31 }+ 4 \lambda_{ 37 }+ 4 \lambda_{ 39 }+ 4 \lambda_{ 41 }+ 4 \lambda_{ 43 }=0 , \end{aligned} \tag{35}\]
\[\begin{aligned} \label{fp72:7} \lambda_{ 8 }-\lambda_{ 9 }+\lambda_{ 10 }-\lambda_{ 11 }+ 3 \lambda_{ 12 }-\lambda_{ 13 }+\lambda_{ 14 }-3 \lambda_{ 15 }+ 2 \lambda_{ 28 }-2 \lambda_{ 29 }+ 2 \lambda_{ 30 }+ 2 \lambda_{ 31 }-2 \lambda_{ 32 }-2 \lambda_{ 33 }-2 \lambda_{ 35 }\notag\\ -2 \lambda_{ 36 }+ 2 \lambda_{ 37 }+ 2 \lambda_{ 38 }-2 \lambda_{ 39 }+ 2 \lambda_{ 40 }-3 \lambda_{ 41 }+ \lambda_{ 42 }-\lambda_{ 43 }+ 3 \lambda_{ 44 }-\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{ 47 }+\lambda_{ 48 }=0 , \end{aligned} \tag{36}\]
\[\begin{aligned} \label{fp72:8} 4 \lambda_{ 10 }+ 4 \lambda_{ 13 }+ 4 \lambda_{ 29 }+ 4 \lambda_{ 31 }+ 4 \lambda_{ 37 }+ 4 \lambda_{ 39 }+ 4 \lambda_{ 43 }+ 4 \lambda_{ 46 }=0 , \end{aligned} \tag{37}\]
\[\begin{aligned} \label{fp72:9} 4 \lambda_{ 12 }+ 4 \lambda_{ 14 }+ 4 \lambda_{ 28 }+ 4 \lambda_{ 30 }+ 4 \lambda_{ 38 }+ 4 \lambda_{ 40 }+ 4 \lambda_{ 42 }+ 4 \lambda_{ 44 }=0 , \end{aligned} \tag{38}\] whereas, \(uv_{5}=0\) if and only if the following equations are satisfied.
\[\begin{aligned} \label{fp82:1} &\lambda_{ 1 }+ 2 \lambda_{ 5 }+ 2 \lambda_{ 6 }-2 \lambda_{ 7 }-\lambda_{ 9 }+ 3 \lambda_{ 11 }+ 3 \lambda_{ 12 }+ 4 \lambda_{ 14 }-3 \lambda_{ 16 }+\lambda_{ 17 }+\lambda_{ 18 }\notag\\&-2 \lambda_{ 19 }+ 2 \lambda_{ 20 }+ 2 \lambda_{ 21 }+\lambda_{ 25 }+\lambda_{ 26 }-3 \lambda_{ 27 } + 4 \lambda_{ 29 }+ 2 \lambda_{ 31 }+ 2 \lambda_{ 32 }+\lambda_{ 34 }+ 2 \lambda_{ 36 }+ 2 \lambda_{ 37 }\notag\\ &+ 4 \lambda_{ 39 }+ 4 \lambda_{ 42 }+ 3 \lambda_{ 44 }+ 3 \lambda_{ 45 }-\lambda_{ 47 }=0 , \end{aligned} \tag{39}\]
\[\begin{aligned} \label{fp82:2} \lambda_{ 2 }+ 2 \lambda_{ 5 }+ 2 \lambda_{ 6 }+\lambda_{ 9 }+ 5 \lambda_{ 11 }+ 3 \lambda_{ 12 }+ 6 \lambda_{ 14 }+ 2 \lambda_{ 17 }+ 2 \lambda_{ 18 }+ 2 \lambda_{ 20 }+ 2 \lambda_{ 21 }+\lambda_{ 24 }+ 2 \lambda_{ 25 }+ 2 \lambda_{ 26 }\notag\\ + 6 \lambda_{ 29 }+ 5 \lambda_{ 31 }+ 4 \lambda_{ 32 }-\lambda_{ 33 }-\lambda_{ 35 }+ 4 \lambda_{ 36 }+ 5 \lambda_{ 37 }+ 6 \lambda_{ 39 }+ 6 \lambda_{ 42 }+ 3 \lambda_{ 44 }+ 5 \lambda_{ 45 }+\lambda_{ 47 }=0 , \end{aligned} \tag{40}\]
\[\begin{aligned} \label{fp82:3} \lambda_{ 3 }+ 3 \lambda_{ 5 }+ 2 \lambda_{ 6 }+\lambda_{ 9 }+ 2 \lambda_{ 10 }+ 3 \lambda_{ 11 }+ 3 \lambda_{ 12 }+ 4 \lambda_{ 14 }+ 2 \lambda_{ 17 }+ 4 \lambda_{ 18 }+ 2 \lambda_{ 20 }+ 3 \lambda_{ 21 }+\lambda_{ 23 }+ 4 \lambda_{ 25 }+ 2 \lambda_{ 26 }\notag\\ + 3 \lambda_{ 29 }+ 4 \lambda_{ 31 }+ 4 \lambda_{ 32 }+ 2 \lambda_{ 33 }+ 2 \lambda_{ 35 }+ 4 \lambda_{ 36 }+ 4 \lambda_{ 37 }+ 3 \lambda_{ 39 }+ 4 \lambda_{ 42 }+ 3 \lambda_{ 44 }+ 3 \lambda_{ 45 }+ 2 \lambda_{ 46 }+\lambda_{ 47 }=0 , \end{aligned} \tag{41}\]
\[\begin{aligned} \label{fp82:4} \lambda_{ 4 }+ 2 \lambda_{ 5 }+ 3 \lambda_{ 6 }+\lambda_{ 9 }+ 2 \lambda_{ 10 }+ 3 \lambda_{ 11 }+\lambda_{ 12 }+ 2 \lambda_{ 14 }+ 4 \lambda_{ 17 }+ 2 \lambda_{ 18 }+ 3 \lambda_{ 20 }+ 2 \lambda_{ 21 }+\lambda_{ 22 }+ 2 \lambda_{ 25 }+ 4 \lambda_{ 26 }\notag\\ + 2 \lambda_{ 29 }+ 4 \lambda_{ 31 }+\lambda_{ 32 }+ 2 \lambda_{ 33 }+ 2 \lambda_{ 35 }+\lambda_{ 36 }+ 4 \lambda_{ 37 }+ 2 \lambda_{ 39 }+ 2 \lambda_{ 42 }+\lambda_{ 44 }+ 3 \lambda_{ 45 }+ 2 \lambda_{ 46 }+\lambda_{ 47 }=0 , \end{aligned} \tag{42}\]
\[\begin{aligned} \label{fp82:5} 4 \lambda_{ 5 }+ 4 \lambda_{ 11 }+ 4 \lambda_{ 18 }+ 4 \lambda_{ 21 }+ 4 \lambda_{ 25 }+ 4 \lambda_{ 31 }+ 4 \lambda_{ 37 }+ 4 \lambda_{ 45 }=0 , \end{aligned} \tag{43}\]
\[\begin{aligned} \label{fp82:6} 4 \lambda_{ 6 }+ 4 \lambda_{ 11 }+ 4 \lambda_{ 14 }+ 4 \lambda_{ 17 }+ 4 \lambda_{ 20 }+ 4 \lambda_{ 26 }+ 4 \lambda_{ 29 }+ 4 \lambda_{ 31 }+ 4 \lambda_{ 37 }+ 4 \lambda_{ 39 }+ 4 \lambda_{ 42 }+ 4 \lambda_{ 45 }=0 , \end{aligned} \tag{44}\]
\[\begin{aligned} \label{fp82:7} \lambda_{ 8 }-\lambda_{ 9 }+\lambda_{ 10 }-\lambda_{ 11 }+ 3 \lambda_{ 12 }-\lambda_{ 13 }+\lambda_{ 14 }-3 \lambda_{ 15 }-2 \lambda_{ 28 }+ 2 \lambda_{ 29 }-2 \lambda_{ 30 }-2 \lambda_{ 31 }+ 2 \lambda_{ 32 }+ 2 \lambda_{ 33 }+ 2 \lambda_{ 35 }\notag\\ + 2 \lambda_{ 36 }-2 \lambda_{ 37 }-2 \lambda_{ 38 }+ 2 \lambda_{ 39 }-2 \lambda_{ 40 }-3 \lambda_{ 41 }+\lambda_{ 42 }-\lambda_{ 43 }+ 3 \lambda_{ 44 }-\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{ 47 }+\lambda_{ 48 }=0 , \end{aligned} \tag{45}\]
\[\begin{aligned} \label{fp82:8} 4 \lambda_{ 10 }+ 4 \lambda_{ 13 }+ 4 \lambda_{ 28 }+ 4 \lambda_{ 33 }+ 4 \lambda_{ 35 }+ 4 \lambda_{ 40 }+ 4 \lambda_{ 43 }+ 4 \lambda_{ 46 }=0 , \end{aligned} \tag{46}\]
\[\begin{aligned} \label{fp82:9} 4 \lambda_{ 12 }+ 4 \lambda_{ 14 }+ 4 \lambda_{ 29 }+ 4 \lambda_{ 32 }+ 4 \lambda_{ 36 }+ 4 \lambda_{ 39 }+ 4 \lambda_{ 42 }+ 4 \lambda_{ 44 }=0 , \end{aligned} \tag{47}\] whereas, \(uv_{6}=0\) if and only if the following equations are satisfied.
\[\begin{aligned} \label{fp92:1} \lambda_{ 1 }+ 2 \lambda_{ 5 }+ 2 \lambda_{ 6 }-\lambda_{ 9 }+ 3 \lambda_{ 12 }+ 3 \lambda_{ 13 }+ 3 \lambda_{ 17 }+ 3 \lambda_{ 18 }+ 6 \lambda_{ 19 }-2 \lambda_{ 20 }-2 \lambda_{ 21 }-3 \lambda_{ 25 }-3 \lambda_{ 26 }\notag\\ + 5 \lambda_{ 27 }+ 2 \lambda_{ 28 }+ 2 \lambda_{ 32 }-\lambda_{ 34 }-2 \lambda_{ 36 }+ 6 \lambda_{ 37 }-2 \lambda_{ 40 }-3 \lambda_{ 43 }-3 \lambda_{ 44 }+ 5 \lambda_{ 45 }+\lambda_{ 47 }=0 , \end{aligned} \tag{48}\]
\[\begin{aligned} \label{fp92:2} \lambda_{ 2 }+ 2 \lambda_{ 5 }+ 2 \lambda_{ 6 }-\lambda_{ 9 }+\lambda_{ 12 }+ 5 \lambda_{ 13 }-2 \lambda_{ 14 }+ 2 \lambda_{ 15 }+ 2 \lambda_{ 17 }+ 2 \lambda_{ 18 }+ 5 \lambda_{ 19 }\notag\\ -2 \lambda_{ 20 }-2 \lambda_{ 21 }-\lambda_{ 24 }-2 \lambda_{ 25 }-2 \lambda_{ 26 }+ 4 \lambda_{ 27 }+ 4 \lambda_{ 28 }-2 \lambda_{ 29 }+ 2 \lambda_{ 30 }+\lambda_{ 33 }\notag\\ -\lambda_{ 35 }+ 5 \lambda_{ 37 }-2 \lambda_{ 38 }+ 2 \lambda_{ 39 }-2 \lambda_{ 41 }+ 2 \lambda_{ 42 }-\lambda_{ 44 }+ 5 \lambda_{ 45 }+\lambda_{ 47 }=0 , \end{aligned} \tag{49}\]
\[\begin{aligned} \label{fp92:3} &\lambda_{ 3 }+ 3 \lambda_{ 5 }+ 2 \lambda_{ 6 }-\lambda_{ 9 }+ 2 \lambda_{ 10 }+\lambda_{ 12 }+ 7 \lambda_{ 13 }+ 2 \lambda_{ 15 }+ 2 \lambda_{ 17 }+ 4 \lambda_{ 18 }+ 6 \lambda_{ 19 }\notag\\ &-2 \lambda_{ 20 }-3 \lambda_{ 21 }-\lambda_{ 23 }-2 \lambda_{ 26 }+ 6 \lambda_{ 27 } + 7 \lambda_{ 28 }-3 \lambda_{ 29 }+ 2 \lambda_{ 30 }+ 2 \lambda_{ 33 }-2 \lambda_{ 35 }\notag\\ &+ 8 \lambda_{ 37 }-2 \lambda_{ 38 }+ 3 \lambda_{ 39 }-2 \lambda_{ 41 }+ 4 \lambda_{ 42 }-\lambda_{ 44 }+ 7 \lambda_{ 45 }-2 \lambda_{ 46 }+\lambda_{ 47 }=0 , \end{aligned} \tag{50}\]
\[\begin{aligned} \label{fp92:4} &\lambda_{ 4 }+ 2 \lambda_{ 5 }+ 3 \lambda_{ 6 }-\lambda_{ 9 }+ 2 \lambda_{ 10 }+ 3 \lambda_{ 12 }+ 5 \lambda_{ 13 }-2 \lambda_{ 14 }+ 4 \lambda_{ 17 }+ 2 \lambda_{ 18 }+ 6 \lambda_{ 19 }\notag\\ &-3 \lambda_{ 20 }-2 \lambda_{ 21 }-\lambda_{ 22 }-2 \lambda_{ 25 }+ 6 \lambda_{ 27 } + 4 \lambda_{ 28 }-2 \lambda_{ 29 }+\lambda_{ 30 }+ 3 \lambda_{ 32 }+ 2 \lambda_{ 33 }-2 \lambda_{ 35 }\notag\\ &-3 \lambda_{ 36 }+ 8 \lambda_{ 37 }-\lambda_{ 38 }+ 2 \lambda_{ 39 }+ 2 \lambda_{ 42 }-3 \lambda_{ 44 }+ 7 \lambda_{ 45 }-2 \lambda_{ 46 }+\lambda_{ 47 }=0 , \end{aligned} \tag{51}\]
\[\begin{aligned} \label{fp92:5} 4 \lambda_{ 5 }+ 4 \lambda_{ 13 }+ 4 \lambda_{ 18 }+ 4 \lambda_{ 19 }+ 4 \lambda_{ 27 }+ 4 \lambda_{ 28 }+ 4 \lambda_{ 37 }+ 4 \lambda_{ 45 }=0 , \end{aligned} \tag{52}\]
\[\begin{aligned} \label{fp92:6} 4 \lambda_{ 6 }+ 4 \lambda_{ 13 }+ 4 \lambda_{ 15 }+ 4 \lambda_{ 17 }+ 4 \lambda_{ 19 }+ 4 \lambda_{ 27 }+ 4 \lambda_{ 28 }+ 4 \lambda_{ 30 }+ 4 \lambda_{ 37 }+ 4 \lambda_{ 39 }+ 4 \lambda_{ 42 }+ 4 \lambda_{ 45 }=0 , \end{aligned} \tag{53}\]
\[\begin{aligned} \label{fp92:7} \lambda_{ 8 }-\lambda_{ 9 }+\lambda_{ 10 }-\lambda_{ 11 }+ 3 \lambda_{ 12 }-\lambda_{ 13 }+\lambda_{ 14 }-3 \lambda_{ 15 }-2 \lambda_{ 28 }+ 2 \lambda_{ 29 }-2 \lambda_{ 30 }-2 \lambda_{ 31 }+ 2 \lambda_{ 32 }+ 2 \lambda_{ 33 }-2 \lambda_{ 35 }\notag\\ -2 \lambda_{ 36 }+ 2 \lambda_{ 37 }+ 2 \lambda_{ 38 }-2 \lambda_{ 39 }+ 2 \lambda_{ 40 }+ 3 \lambda_{ 41 }-\lambda_{ 42 }+\lambda_{ 43 }-3 \lambda_{ 44 }+\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{ 47 }-\lambda_{ 48 }=0 , \end{aligned} \tag{54}\]
\[\begin{aligned} \label{fp92:8} 4 \lambda_{ 10 }+ 4 \lambda_{ 13 }+ 4 \lambda_{ 28 }+ 4 \lambda_{ 33 }+ 4 \lambda_{ 37 }+ 4 \lambda_{ 39 }+ 4 \lambda_{ 42 }+ 4 \lambda_{ 45 }=0 , \end{aligned} \tag{55}\]
\[\begin{aligned} \label{fp92:9} 4 \lambda_{ 12 }+ 4 \lambda_{ 14 }+ 4 \lambda_{ 29 }+ 4 \lambda_{ 32 }+ 4 \lambda_{ 38 }+ 4 \lambda_{ 40 }+ 4 \lambda_{ 41 }+ 4 \lambda_{ 43 }=0. \end{aligned} \tag{56}\]
Suppose that \(\mathscr{I}\) denotes the corresponding ideal in any possible case of \((i)\), \((ii)\), \((iii)\) and \((iv)\). We claim that \(d(\mathscr{I})=2\). For this, consider the corresponding ideal \(\mathscr{I}\) of \(\mathscr{F}\overrightarrow{S_{3}}\) containing the elements of the form \(u=\sum\limits_{i=1}^{48} \lambda_{i}g_{i}\) satisfying \(ux=0\), for all \(x\in\Lambda\), where \(\Lambda\) is a subset of the set \(\{v_{3}, v_{4}, v_{5}, v_{6}\}\), which depends upon the case under consideration. This infers that the coefficients in the presentation of \(u\) must satisfy suitable equations from the set of equation (21) to (56), corresponding to \(ux=0\), for all \(x\in\Lambda\). Clearly, \(\mathscr{I}\) can not contain any element of weight \(1\) and so \(d(\mathscr{I})\geq 2\). Further, an element \(u\) with the smallest weight \(2\) in \(\mathscr{I}\) is \(u=g_{1}+g_{34}\) if \(\Lambda\subseteq\{v_{3},v_{6}\}\), while \(u=g_{1}-g_{34}\) if \(\Lambda\subseteq\{v_{4},v_{5}\}\). Thus, we conclude that \(d(\mathscr{I})=2\). Next, for the dimension of \(\mathscr{I}\), observe that all the equation (21) to (56) are linearly independent, therefore the dimension of ideal \(\mathscr{I}\) is \(48-4|\Lambda|\). This completes the proof. ◻
We summarize some other results concerning group codes in \(\mathscr{F}[\overrightarrow{S_{3}}]\) in the upcoming theorems. In fact these results can be proved by using similar techniques with necessary variation that we have used in our previous results of this section.
Theorem 4.8. Let \(e_{1}\), \(e_{2}\), \(e_{3}\), \(e_{4}\) and \(v_{1}\), \(v_{2}\), \(v_{3}\), \(v_{4}\), \(v_{5}\), \(v_{6}\) be linear and nonlinear idempotents in \(\mathscr{F}[\overrightarrow{S_{3}}]\). If \(X_{i}\) and \(Z_{j}\) are subsets of order \(i\) and \(j\) of the sets \(\{e_{1}, e_{4}\}\) and \(\{v_{4}, v_{5}\}\) respectively, then
\((i)\) \(d(\mathscr{I}_{X_{1}\cup Z_{1}})=2\) and \(dim(\mathscr{I}_{X_{1}\cup Z_{1}})=38\).
\((ii)\) \(d(\mathscr{I}_{X_{1}\cup Z_{2}})=2\) and \(dim(\mathscr{I}_{X_{1}\cup Z_{2}})=29\).
\((iii)\) \(d(\mathscr{I}_{X_{2}\cup Z_{1}})=2\) and \(dim(\mathscr{I}_{X_{2}\cup Z_{1}})=37\).
\((iv)\) \(d(\mathscr{I}_{X_{2}\cup Z_{2}})=2\) and \(dim(\mathscr{I}_{X_{2}\cup Z_{2}})=28\).
\((v)\) \(d(\mathscr{I}_{\{v_{2}\}\cup Z_{1}})=2\) and \(dim(\mathscr{I}_{\{v_{2}\}\cup Z_{1}})=35\).
\((vi)\) \(d(\mathscr{I}_{\{v_{2}\}\cup Z_{2}})=2\) and \(dim(\mathscr{I}_{\{v_{2}\}\cup Z_{2}})=26\).
\((vii)\) \(d(\mathscr{I}_{X_{1}\cup \{v_{1}\}\cup Z_{1}})=2\) and \(dim(\mathscr{I}_{X_{1}\cup \{v_{1}\}\cup Z_{1}})=34\).
\((viii)\) \(d(\mathscr{I}_{X_{1}\cup \{v_{1}\}\cup Z_{2}})=2\) and \(dim(\mathscr{I}_{X_{1}\cup \{v_{1}\}\cup Z_{2}})=25\).
\((ix)\) \(d(\mathscr{I}_{X_{2}\cup \{v_{1}\}\cup Z_{1}})=2\) and \(dim(\mathscr{I}_{X_{2}\cup \{v_{1}\}\cup Z_{1}})=33\).
\((x)\) \(d(\mathscr{I}_{X_{2}\cup \{v_{1}\}\cup Z_{2}})=2\) and \(dim(\mathscr{I}_{X_{2}\cup \{v_{1}\}\cup Z_{2}})=24\).
Theorem 4.9. Let \(e_{1}\), \(e_{2}\), \(e_{3}\), \(e_{4}\) and \(v_{1}\), \(v_{2}\), \(v_{3}\), \(v_{4}\), \(v_{5}\), \(v_{6}\) be linear and nonlinear idempotents in \(\mathscr{F}[\overrightarrow{S_{3}}]\). If \(X_{i}\) and \(Z_{j}\) are subsets of order \(i\) and \(j\) of the sets \(\{e_{2}, e_{3}\}\) and \(\{v_{3}, v_{6}\}\) respectively, then
\((i)\) \(\mathscr{I}_{X_{1}\cup Z_{1}}\) is \((48, 38, 2)\) group code.
\((ii)\) \(\mathscr{I}_{X_{1}\cup Z_{2}}\) is \((48, 29, 2)\) group code.
\((iii)\) \(\mathscr{I}_{X_{2}\cup Z_{1}}\) is \((48, 37, 2)\) group code.
\((iv)\) \(\mathscr{I}_{X_{2}\cup Z_{2}}\) is \((48, 28, 2)\) group code.
\((v)\) \(\mathscr{I}_{\{v_{2}\}\cup Z_{1}}\) is \((48, 35, 2)\) group code.
\((vi)\) \(\mathscr{I}_{\{v_{2}\}\cup Z_{2}}\) is \((48, 26, 2)\) group code.
\((vii)\) \(\mathscr{I}_{X_{1}\cup \{v_{2}\}\cup Z_{1}}\) is \((48, 34, 2)\) group code.
\((viii)\) \(\mathscr{I}_{X_{1}\cup \{v_{2}\}\cup Z_{2}}\) is \((48, 25, 2)\) group code.
\((ix)\) \(\mathscr{I}_{X_{2}\cup \{v_{2}\}\cup Z_{1}}\) is \((48, 33, 2)\) group code.
\((x)\) \(\mathscr{I}_{X_{2}\cup \{v_{2}\}\cup Z_{2}}\) is \((48, 24, 2)\) group code.
This paper completely characterized the four families of group codes
namely; \((2^{n}n!, 2^{n}n!-1, 2)\)
(MDS group code), \((2^{n}n!, 2^{n}n!-2,
2)\), \((2^{n}n!, 2^{n}n!-3,
2)\) and \((2^{n}n!, 2^{n}n!-4,
2)\) group code which are generated by the linear idempotents
corresponding to the linear characters of \(\overrightarrow{S_{n}}\) in the
hyperoctahedral group algebra \(\mathscr{F}[\overrightarrow{S_{n}}]\) for
every \(n\) and thereby established two
families of group codes \((2^{n-1}n!,
2^{n-1}n!-1, 2)\) (MDS group code) and \((2^{n-1}n!, 2^{n-1}n!-2, 2)\) group code in
\(\mathscr{F}[\overrightarrow{A_{n}}]\)
generated by linear idempotents corresponding to linear characters of
\(\overrightarrow{A_{n}}\), for every
\(n\). In addition, as the order and
the number of nonlinear characters of the group \(\overrightarrow{S_{n}}\) increases very
rapidly with \(n\), therefore the
problem of complete characterization of group code generated by
nonlinear idempotents for arbitrary \(n\), in \(\mathscr{F}[\overrightarrow{S_{n}}]\) is
very lengthy to compute, in general. Moreover, we also attempted to
explore various families of group codes of length \(48\) having minimum distance \(2\), generated by linear and nonlinear
idempotents \(\mathscr{F}[\overrightarrow{S_{3}}]\) of
hyperoctahedral group of small order.
\(uv_{2}=\frac{1}{48}\big[ 2 \lambda_{ 1
}-\lambda_{ 8 }-\lambda_{ 9 }-\lambda_{ 10 }-\lambda_{ 11 }-\lambda_{ 12
}-\lambda_{ 13 }-\lambda_{ 14 }-\lambda_{ 15 }-2 \lambda_{ 16 }-2
\lambda_{ 17 }-2 \lambda_{ 18 }+ 2 \lambda_{ 25 }+ 2 \lambda_{ 26 }+ 2
\lambda_{ 27 }-2 \lambda_{ 34 }+\lambda_{ 41 }+\lambda_{ 42 }+\lambda_{
43 }+\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{ 47
}+\lambda_{ 48 }\big]\big(g_{ 1 }+g_{ 25 }+g_{ 26 }+g_{ 27 }-g_{ 16
}-g_{ 17 }-g_{ 18 }-g_{ 34 }\big)+%1
\big[ 2 \lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{ 5
}-\lambda_{ 6 }+ 2 \lambda_{ 7 }-2 \lambda_{ 19 }+\lambda_{ 20
}+\lambda_{ 21 }+\lambda_{ 22 }+\lambda_{ 23 }-2 \lambda_{ 24
}+\lambda_{ 28 }+\lambda_{ 29 }+\lambda_{ 30 }-2 \lambda_{ 31
}+\lambda_{ 32 }-2 \lambda_{ 33 }+ 2 \lambda_{ 35 }-\lambda_{ 36 }+ 2
\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39 }-\lambda_{ 40 }\big]\big(g_{
2 }+g_{ 7 }+g_{ 35 }+g_{ 37 }-g_{ 19 }-g_{ 24 }-g_{ 31 }-g_{ 33}\big)+%2
\big[-\lambda_{ 2 }+ 2 \lambda_{ 3 }-\lambda_{ 4 }+ 2 \lambda_{ 5
}-\lambda_{ 6 }-\lambda_{ 7 }+\lambda_{ 19 }+\lambda_{ 20 }-2 \lambda_{
21 }+\lambda_{ 22 }-2 \lambda_{ 23 }+\lambda_{ 24 }-2 \lambda_{ 28 }-2
\lambda_{ 29 }+\lambda_{ 30 }+\lambda_{ 31 }+\lambda_{ 32 }+\lambda_{ 33
}-\lambda_{ 35 }-\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }+ 2
\lambda_{ 39 }+ 2 \lambda_{ 40 }\big]\big(g_{ 3 }+g_{ 5 }+g_{ 39 }+g_{
40 }-g_{ 21 }-g_{ 23 }-g_{ 28 }-g_{ 29 }\big)+%3
\big[-\lambda_{ 2 }-\lambda_{ 3 }+ 2 \lambda_{ 4 }-\lambda_{ 5 }+ 2
\lambda_{ 6 }-\lambda_{ 7 }+\lambda_{ 19 }-2 \lambda_{ 20 }+\lambda_{ 21
}-2 \lambda_{ 22 }+\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 28
}+\lambda_{ 29 }-2 \lambda_{ 30 }+\lambda_{ 31 }-2 \lambda_{ 32
}+\lambda_{ 33 }-\lambda_{ 35 }+ 2 \lambda_{ 36 }-\lambda_{ 37 }+ 2
\lambda_{ 38 }-\lambda_{ 39 }-\lambda_{ 40 }\big]\big(g_{ 4 }+g_{ 6
}+g_{ 36 }+g_{ 38 }-g_{ 20 }-g_{ 22 }-g_{ 30 }-g_{ 32}\big)+%4
\big[-\lambda_{ 1 }+ 2 \lambda_{ 8 }-\lambda_{ 9 }+ 2 \lambda_{ 10
}-\lambda_{ 11 }-\lambda_{ 12 }-\lambda_{ 13 }+ 2 \lambda_{ 14 }+ 2
\lambda_{ 15 }+\lambda_{ 16 }+\lambda_{ 17 }+\lambda_{ 18 }-\lambda_{ 25
}-\lambda_{ 26 }-\lambda_{ 27 }+\lambda_{ 34 }-2 \lambda_{ 41 }-2
\lambda_{ 42 }+\lambda_{ 43 }+\lambda_{ 44 }+\lambda_{ 45 }-2 \lambda_{
46 }+\lambda_{ 47 }-2 \lambda_{ 48 }\big]\big(g_{ 8 }+g_{ 10 }+g_{ 14
}+g_{ 15 }-g_{ 41 }-g_{ 42 }-g_{ 46 }-g_{ 48 }\big)+%5
\big[-\lambda_{ 1 }-\lambda_{ 8 }+ 2 \lambda_{ 9 }-\lambda_{ 10 }+ 2
\lambda_{ 11 }+ 2 \lambda_{ 12 }+ 2 \lambda_{ 13 }-\lambda_{ 14
}-\lambda_{ 15 }+\lambda_{ 16 }+\lambda_{ 17 }+\lambda_{ 18 }-\lambda_{
25 }-\lambda_{ 26 }-\lambda_{ 27 }+\lambda_{ 34 }+\lambda_{ 41
}+\lambda_{ 42 }-2 \lambda_{ 43 }-2 \lambda_{ 44 }-2 \lambda_{ 45
}+\lambda_{ 46 }-2 \lambda_{ 47 }+\lambda_{ 48 }\big]\big(g_{ 9 }+g_{ 11
}+g_{ 12}+g_{ 13 }-g_{ 43 }-g_{ 44 }-g_{ 45 }-g_{ 47
}\big).\)
\(uv_{3}=\frac{1}{48}\big[ 3 \lambda_{ 1
}+\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{ 5 }+\lambda_{ 6
}+\lambda_{ 7 }+\lambda_{ 16 }+\lambda_{ 17 }+\lambda_{ 18 }-\lambda_{
19 }-\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{ 22 }-\lambda_{ 23
}-\lambda_{ 24 }-\lambda_{ 25 }-\lambda_{ 26 }-\lambda_{ 27 }+\lambda_{
28 }+\lambda_{ 29 }+\lambda_{ 30 }+\lambda_{ 31 }+\lambda_{ 32
}+\lambda_{ 33 }-3 \lambda_{ 34 }-\lambda_{ 35 }-\lambda_{ 36
}-\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39 }-\lambda_{ 40 }\big]g_{ 1
}+% 1
\big[\lambda_{ 1 }+ 3 \lambda_{ 2 }-\lambda_{ 7 }+\lambda_{ 8
}+\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }-\lambda_{
13 }-\lambda_{ 14 }-\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17
}+\lambda_{ 18 }+\lambda_{ 19 }-3 \lambda_{ 24 }-\lambda_{ 25
}-\lambda_{ 26 }+\lambda_{ 27 }+\lambda_{ 31 }+\lambda_{ 33 }-\lambda_{
34 }-\lambda_{ 35 }-\lambda_{ 37 }+\lambda_{ 41 }+\lambda_{ 42
}+\lambda_{ 43 }+\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{
47 }-\lambda_{ 48 }\big]g_{ 2 }+% 2
\big[\lambda_{ 1 }+ 3 \lambda_{ 3 }-\lambda_{ 5 }+\lambda_{ 8
}+\lambda_{ 9 }-\lambda_{ 10 }-\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{
13 }+\lambda_{ 14 }-\lambda_{ 15 }+\lambda_{ 16 }+\lambda_{ 17
}-\lambda_{ 18 }+\lambda_{ 21 }-3 \lambda_{ 23 }+\lambda_{ 25
}-\lambda_{ 26 }-\lambda_{ 27 }+\lambda_{ 28 }+\lambda_{ 29 }-\lambda_{
34 }-\lambda_{ 39 }-\lambda_{ 40 }+\lambda_{ 41 }-\lambda_{ 42
}-\lambda_{ 43 }+\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{
47 }-\lambda_{ 48 }\big]g_{ 3 }+% 3
\big[\lambda_{ 1 }+ 3 \lambda_{ 4 }-\lambda_{ 6 }+\lambda_{ 8
}+\lambda_{ 9 }-\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{
13 }-\lambda_{ 14 }+\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17
}+\lambda_{ 18 }+\lambda_{ 20 }-3 \lambda_{ 22 }-\lambda_{ 25
}+\lambda_{ 26 }-\lambda_{ 27 }+\lambda_{ 30 }+\lambda_{ 32 }-\lambda_{
34 }-\lambda_{ 36 }-\lambda_{ 38 }-\lambda_{ 41 }+\lambda_{ 42
}+\lambda_{ 43 }-\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{
47 }-\lambda_{ 48 }\big]g_{ 4 }+% 4
\big[\lambda_{ 1 }-\lambda_{ 3 }+ 3 \lambda_{ 5 }-\lambda_{ 8
}-\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{
13 }-\lambda_{ 14 }+\lambda_{ 15 }+\lambda_{ 16 }+\lambda_{ 17
}-\lambda_{ 18 }-3 \lambda_{ 21 }+\lambda_{ 23 }+\lambda_{ 25
}-\lambda_{ 26 }-\lambda_{ 27 }+\lambda_{ 28 }+\lambda_{ 29 }-\lambda_{
34 }-\lambda_{ 39 }-\lambda_{ 40 }-\lambda_{ 41 }+\lambda_{ 42
}+\lambda_{ 43 }-\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{
47 }+\lambda_{ 48 }\big]g_{ 5 }+% 5
\big[\lambda_{ 1 }-\lambda_{ 4 }+ 3 \lambda_{ 6 }-\lambda_{ 8
}-\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{
13 }+\lambda_{ 14 }-\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17
}+\lambda_{ 18 }-3 \lambda_{ 20 }+\lambda_{ 22 }-\lambda_{ 25
}+\lambda_{ 26 }-\lambda_{ 27 }+\lambda_{ 30 }+\lambda_{ 32 }-\lambda_{
34 }-\lambda_{ 36 }-\lambda_{ 38 }+\lambda_{ 41 }-\lambda_{ 42
}-\lambda_{ 43 }+\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{
47 }+\lambda_{ 48 }\big]g_{ 6 }+% 6
\big[\lambda_{ 1 }-\lambda_{ 2 }+ 3 \lambda_{ 7 }-\lambda_{ 8
}-\lambda_{ 9 }-\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }+\lambda_{
13 }+\lambda_{ 14 }+\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17
}+\lambda_{ 18 }-3 \lambda_{ 19 }+\lambda_{ 24 }-\lambda_{ 25
}-\lambda_{ 26 }+\lambda_{ 27 }+\lambda_{ 31 }+\lambda_{ 33 }-\lambda_{
34 }-\lambda_{ 35 }-\lambda_{ 37 }-\lambda_{ 41 }-\lambda_{ 42
}-\lambda_{ 43 }-\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{
47 }+\lambda_{ 48 }\big]g_{ 7}+% 7
\big[\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }-\lambda_{ 5 }-\lambda_{ 6
}-\lambda_{ 7 }+ 3 \lambda_{ 8 }-\lambda_{ 10 }-\lambda_{ 14 }-\lambda_{
15 }+\lambda_{ 19 }+\lambda_{ 20 }+\lambda_{ 21 }-\lambda_{ 22
}-\lambda_{ 23 }-\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29 }-\lambda_{
30 }+\lambda_{ 31 }+\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{ 35
}-\lambda_{ 36 }-\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39 }-\lambda_{
40 }+\lambda_{ 41 }+\lambda_{ 42 }+\lambda_{ 46 }-3 \lambda_{ 48
}\big]g_{ 8 }+% 8
\big[\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }-\lambda_{ 5 }-\lambda_{ 6
}-\lambda_{ 7 }+ 3 \lambda_{ 9 }-\lambda_{ 11 }-\lambda_{ 12 }-\lambda_{
13 }+\lambda_{ 19 }+\lambda_{ 20 }+\lambda_{ 21 }-\lambda_{ 22
}-\lambda_{ 23 }-\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29 }+\lambda_{
30 }-\lambda_{ 31 }-\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{ 35
}+\lambda_{ 36 }+\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39 }+\lambda_{
40 }+\lambda_{ 43 }+\lambda_{ 44 }+\lambda_{ 45 }-3 \lambda_{ 47
}\big]g_{ 9 }+% 9
\big[\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }+\lambda_{ 5 }+\lambda_{ 6
}-\lambda_{ 7 }-\lambda_{ 8 }+ 3 \lambda_{ 10 }-\lambda_{ 14 }-\lambda_{
15 }+\lambda_{ 19 }-\lambda_{ 20 }-\lambda_{ 21 }+\lambda_{ 22
}+\lambda_{ 23 }-\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29 }+\lambda_{
30 }+\lambda_{ 31 }-\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{ 35
}+\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39 }+\lambda_{
40 }+\lambda_{ 41 }+\lambda_{ 42 }-3 \lambda_{ 46 }+\lambda_{ 48
}\big]g_{ 10}+% 10
\big[\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }+\lambda_{ 5 }+\lambda_{ 6
}-\lambda_{ 7 }-\lambda_{ 9 }+ 3 \lambda_{ 11 }-\lambda_{ 12 }-\lambda_{
13 }+\lambda_{ 19 }-\lambda_{ 20 }-\lambda_{ 21 }+\lambda_{ 22
}+\lambda_{ 23 }-\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29 }-\lambda_{
30 }-\lambda_{ 31 }+\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{ 35
}-\lambda_{ 36 }+\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39 }-\lambda_{
40 }+\lambda_{ 43 }+\lambda_{ 44 }-3 \lambda_{ 45 }+\lambda_{ 47
}\big]g_{ 11 }+% 11
\big[-\lambda_{ 2 }-\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{ 5 }-\lambda_{
6 }+\lambda_{ 7 }-\lambda_{ 9 }-\lambda_{ 11 }+ 3 \lambda_{ 12
}-\lambda_{ 13 }-\lambda_{ 19 }+\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{
22 }+\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29
}+\lambda_{ 30 }+\lambda_{ 31 }-\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{
35 }+\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }+\lambda_{ 39
}-\lambda_{ 40 }+\lambda_{ 43 }-3 \lambda_{ 44 }+\lambda_{ 45
}+\lambda_{ 47 }\big]g_{ 12 }+% 12
\big[-\lambda_{ 2 }+\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{ 5 }+\lambda_{
6 }+\lambda_{ 7 }-\lambda_{ 9 }-\lambda_{ 11 }-\lambda_{ 12 }+ 3
\lambda_{ 13 }-\lambda_{ 19 }-\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{ 22
}-\lambda_{ 23 }+\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29 }-\lambda_{
30 }+\lambda_{ 31 }+\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{ 35
}-\lambda_{ 36 }-\lambda_{ 37 }+\lambda_{ 38 }-\lambda_{ 39 }+\lambda_{
40 }-3 \lambda_{ 43 }+\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 47
}\big]g_{ 13 }+% 13
\big[-\lambda_{ 2 }+\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{ 5 }+\lambda_{
6 }+\lambda_{ 7 }-\lambda_{ 8 }-\lambda_{ 10 }+ 3 \lambda_{ 14
}-\lambda_{ 15 }-\lambda_{ 19 }-\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{
22 }-\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29
}+\lambda_{ 30 }-\lambda_{ 31 }-\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{
35 }+\lambda_{ 36 }+\lambda_{ 37 }-\lambda_{ 38 }+\lambda_{ 39
}-\lambda_{ 40 }+\lambda_{ 41 }-3 \lambda_{ 42 }+\lambda_{ 46
}+\lambda_{ 48 }\big]g_{ 14 }+% 14
\big[-\lambda_{ 2 }-\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{ 5 }-\lambda_{
6 }+\lambda_{ 7 }-\lambda_{ 8 }-\lambda_{ 10 }-\lambda_{ 14 }+ 3
\lambda_{ 15 }-\lambda_{ 19 }+\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{ 22
}+\lambda_{ 23 }+\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29 }-\lambda_{
30 }-\lambda_{ 31 }+\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{ 35
}-\lambda_{ 36 }+\lambda_{ 37 }+\lambda_{ 38 }-\lambda_{ 39 }+\lambda_{
40 }-3 \lambda_{ 41 }+\lambda_{ 42 }+\lambda_{ 46 }+\lambda_{ 48
}\big]g_{ 15 }+% 15
\big[\lambda_{ 1 }-\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{ 5
}+\lambda_{ 6 }-\lambda_{ 7 }+ 3 \lambda_{ 16 }-\lambda_{ 17 }-\lambda_{
18 }+\lambda_{ 19 }-\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{ 22
}-\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 25 }+\lambda_{ 26 }-3
\lambda_{ 27 }+\lambda_{ 28 }+\lambda_{ 29 }+\lambda_{ 30 }-\lambda_{ 31
}+\lambda_{ 32 }-\lambda_{ 33 }-\lambda_{ 34 }+\lambda_{ 35 }-\lambda_{
36 }+\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39 }-\lambda_{ 40 }\big]g_{
16 }+% 16
\big[\lambda_{ 1 }+\lambda_{ 2 }+\lambda_{ 3 }-\lambda_{ 4 }+\lambda_{ 5
}-\lambda_{ 6 }+\lambda_{ 7 }-\lambda_{ 16 }+ 3 \lambda_{ 17 }-\lambda_{
18 }-\lambda_{ 19 }+\lambda_{ 20 }-\lambda_{ 21 }+\lambda_{ 22
}-\lambda_{ 23 }-\lambda_{ 24 }+\lambda_{ 25 }-3 \lambda_{ 26
}+\lambda_{ 27 }+\lambda_{ 28 }+\lambda_{ 29 }-\lambda_{ 30 }+\lambda_{
31 }-\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{ 34 }-\lambda_{ 35
}+\lambda_{ 36 }-\lambda_{ 37 }+\lambda_{ 38 }-\lambda_{ 39 }-\lambda_{
40 }\big]g_{ 17 }+% 17
\big[\lambda_{ 1 }+\lambda_{ 2 }-\lambda_{ 3 }+\lambda_{ 4 }-\lambda_{ 5
}+\lambda_{ 6 }+\lambda_{ 7 }-\lambda_{ 16 }-\lambda_{ 17 }+ 3 \lambda_{
18 }-\lambda_{ 19 }-\lambda_{ 20 }+\lambda_{ 21 }-\lambda_{ 22
}+\lambda_{ 23 }-\lambda_{ 24 }-3 \lambda_{ 25 }+\lambda_{ 26
}+\lambda_{ 27 }-\lambda_{ 28 }-\lambda_{ 29 }+\lambda_{ 30 }+\lambda_{
31 }+\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{ 34 }-\lambda_{ 35
}-\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }+\lambda_{ 39 }+\lambda_{
40 }\big]g_{ 18 }+% 18
\big[-\lambda_{ 1 }+\lambda_{ 2 }-3 \lambda_{ 7 }+\lambda_{ 8
}+\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }-\lambda_{
13 }-\lambda_{ 14 }-\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17
}-\lambda_{ 18 }+ 3 \lambda_{ 19 }-\lambda_{ 24 }+\lambda_{ 25
}+\lambda_{ 26 }-\lambda_{ 27 }-\lambda_{ 31 }-\lambda_{ 33 }+\lambda_{
34 }+\lambda_{ 35 }+\lambda_{ 37 }+\lambda_{ 41 }+\lambda_{ 42
}+\lambda_{ 43 }+\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{
47 }-\lambda_{ 48 }\big]g_{ 19}+% 19
\big[-\lambda_{ 1 }+\lambda_{ 4 }-3 \lambda_{ 6 }+\lambda_{ 8
}+\lambda_{ 9 }-\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{
13 }-\lambda_{ 14 }+\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17
}-\lambda_{ 18 }+ 3 \lambda_{ 20 }-\lambda_{ 22 }+\lambda_{ 25
}-\lambda_{ 26 }+\lambda_{ 27 }-\lambda_{ 30 }-\lambda_{ 32 }+\lambda_{
34 }+\lambda_{ 36 }+\lambda_{ 38 }-\lambda_{ 41 }+\lambda_{ 42
}+\lambda_{ 43 }-\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{
47 }-\lambda_{ 48 }\big]g_{ 20 }+% 20
\big[-\lambda_{ 1 }+\lambda_{ 3 }-3 \lambda_{ 5 }+\lambda_{ 8
}+\lambda_{ 9 }-\lambda_{ 10 }-\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{
13 }+\lambda_{ 14 }-\lambda_{ 15 }-\lambda_{ 16 }-\lambda_{ 17
}+\lambda_{ 18 }+ 3 \lambda_{ 21 }-\lambda_{ 23 }-\lambda_{ 25
}+\lambda_{ 26 }+\lambda_{ 27 }-\lambda_{ 28 }-\lambda_{ 29 }+\lambda_{
34 }+\lambda_{ 39 }+\lambda_{ 40 }+\lambda_{ 41 }-\lambda_{ 42
}-\lambda_{ 43 }+\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{
47 }-\lambda_{ 48 }\big]g_{ 21 }+% 21
\big[-\lambda_{ 1 }-3 \lambda_{ 4 }+\lambda_{ 6 }-\lambda_{ 8
}-\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{
13 }+\lambda_{ 14 }-\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17
}-\lambda_{ 18 }-\lambda_{ 20 }+ 3 \lambda_{ 22 }+\lambda_{ 25
}-\lambda_{ 26 }+\lambda_{ 27 }-\lambda_{ 30 }-\lambda_{ 32 }+\lambda_{
34 }+\lambda_{ 36 }+\lambda_{ 38 }+\lambda_{ 41 }-\lambda_{ 42
}-\lambda_{ 43 }+\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{
47 }+\lambda_{ 48 }\big]g_{ 22}+% 22
\big[-\lambda_{ 1 }-3 \lambda_{ 3 }+\lambda_{ 5 }-\lambda_{ 8
}-\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{
13 }-\lambda_{ 14 }+\lambda_{ 15 }-\lambda_{ 16 }-\lambda_{ 17
}+\lambda_{ 18 }-\lambda_{ 21 }+ 3 \lambda_{ 23 }-\lambda_{ 25
}+\lambda_{ 26 }+\lambda_{ 27 }-\lambda_{ 28 }-\lambda_{ 29 }+\lambda_{
34 }+\lambda_{ 39 }+\lambda_{ 40 }-\lambda_{ 41 }+\lambda_{ 42
}+\lambda_{ 43 }-\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{
47 }+\lambda_{ 48 }\big]g_{ 23 }+% 23
\big[-\lambda_{ 1 }-3 \lambda_{ 2 }+\lambda_{ 7 }-\lambda_{ 8
}-\lambda_{ 9 }-\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }+\lambda_{
13 }+\lambda_{ 14 }+\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17
}-\lambda_{ 18 }-\lambda_{ 19 }+ 3 \lambda_{ 24 }+\lambda_{ 25
}+\lambda_{ 26 }-\lambda_{ 27 }-\lambda_{ 31 }-\lambda_{ 33 }+\lambda_{
34 }+\lambda_{ 35 }+\lambda_{ 37 }-\lambda_{ 41 }-\lambda_{ 42
}-\lambda_{ 43 }-\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{
47 }+\lambda_{ 48 }\big]g_{ 24 }+% 24
\big[-\lambda_{ 1 }-\lambda_{ 2 }+\lambda_{ 3 }-\lambda_{ 4 }+\lambda_{
5 }-\lambda_{ 6 }-\lambda_{ 7 }+\lambda_{ 16 }+\lambda_{ 17 }-3
\lambda_{ 18 }+\lambda_{ 19 }+\lambda_{ 20 }-\lambda_{ 21 }+\lambda_{ 22
}-\lambda_{ 23 }+\lambda_{ 24 }+ 3 \lambda_{ 25 }-\lambda_{ 26
}-\lambda_{ 27 }+\lambda_{ 28 }+\lambda_{ 29 }-\lambda_{ 30 }-\lambda_{
31 }-\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{ 34 }+\lambda_{ 35
}+\lambda_{ 36 }+\lambda_{ 37 }+\lambda_{ 38 }-\lambda_{ 39 }-\lambda_{
40 }\big]g_{ 25 }+% 25
\big[-\lambda_{ 1 }-\lambda_{ 2 }-\lambda_{ 3 }+\lambda_{ 4 }-\lambda_{
5 }+\lambda_{ 6 }-\lambda_{ 7 }+\lambda_{ 16 }-3 \lambda_{ 17
}+\lambda_{ 18 }+\lambda_{ 19 }-\lambda_{ 20 }+\lambda_{ 21 }-\lambda_{
22 }+\lambda_{ 23 }+\lambda_{ 24 }-\lambda_{ 25 }+ 3 \lambda_{ 26
}-\lambda_{ 27 }-\lambda_{ 28 }-\lambda_{ 29 }+\lambda_{ 30 }-\lambda_{
31 }+\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{ 34 }+\lambda_{ 35
}-\lambda_{ 36 }+\lambda_{ 37 }-\lambda_{ 38 }+\lambda_{ 39 }+\lambda_{
40 }\big]g_{ 26 }+% 26
\big[-\lambda_{ 1 }+\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{
5 }-\lambda_{ 6 }+\lambda_{ 7 }-3 \lambda_{ 16 }+\lambda_{ 17
}+\lambda_{ 18 }-\lambda_{ 19 }+\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{
22 }+\lambda_{ 23 }-\lambda_{ 24 }-\lambda_{ 25 }-\lambda_{ 26 }+ 3
\lambda_{ 27 }-\lambda_{ 28 }-\lambda_{ 29 }-\lambda_{ 30 }+\lambda_{ 31
}-\lambda_{ 32 }+\lambda_{ 33 }+\lambda_{ 34 }-\lambda_{ 35 }+\lambda_{
36 }-\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39 }+\lambda_{ 40 }\big]g_{
27 }+% 27
\big[\lambda_{ 1 }+\lambda_{ 3 }+\lambda_{ 5 }+\lambda_{ 8 }-\lambda_{ 9
}-\lambda_{ 10 }+\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{ 13 }+\lambda_{
14 }-\lambda_{ 15 }+\lambda_{ 16 }+\lambda_{ 17 }-\lambda_{ 18
}-\lambda_{ 21 }-\lambda_{ 23 }+\lambda_{ 25 }-\lambda_{ 26 }-\lambda_{
27 }+ 3 \lambda_{ 28 }-\lambda_{ 29 }-\lambda_{ 34 }+\lambda_{ 39 }-3
\lambda_{ 40 }+\lambda_{ 41 }-\lambda_{ 42 }+\lambda_{ 43 }-\lambda_{ 44
}-\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{ 47 }-\lambda_{ 48 }\big]g_{ 28
}+% 28
\big[\lambda_{ 1 }+\lambda_{ 3 }+\lambda_{ 5 }-\lambda_{ 8 }+\lambda_{ 9
}+\lambda_{ 10 }-\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{ 13 }-\lambda_{
14 }+\lambda_{ 15 }+\lambda_{ 16 }+\lambda_{ 17 }-\lambda_{ 18
}-\lambda_{ 21 }-\lambda_{ 23 }+\lambda_{ 25 }-\lambda_{ 26 }-\lambda_{
27 }-\lambda_{ 28 }+ 3 \lambda_{ 29 }-\lambda_{ 34 }-3 \lambda_{ 39
}+\lambda_{ 40 }-\lambda_{ 41 }+\lambda_{ 42 }-\lambda_{ 43 }+\lambda_{
44 }+\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{ 47 }+\lambda_{ 48 }\big]g_{
29 }+% 29
\big[\lambda_{ 1 }+\lambda_{ 4 }+\lambda_{ 6 }-\lambda_{ 8 }+\lambda_{ 9
}+\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{ 13 }+\lambda_{
14 }-\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17 }+\lambda_{ 18
}-\lambda_{ 20 }-\lambda_{ 22 }-\lambda_{ 25 }+\lambda_{ 26 }-\lambda_{
27 }+ 3 \lambda_{ 30 }-\lambda_{ 32 }-\lambda_{ 34 }+\lambda_{ 36 }-3
\lambda_{ 38 }+\lambda_{ 41 }-\lambda_{ 42 }+\lambda_{ 43 }-\lambda_{ 44
}+\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{ 47 }+\lambda_{ 48 }\big]g_{ 30
}+% 30
\big[\lambda_{ 1 }+\lambda_{ 2 }+\lambda_{ 7 }+\lambda_{ 8 }-\lambda_{ 9
}+\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }+\lambda_{ 13 }-\lambda_{
14 }-\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17 }+\lambda_{ 18
}-\lambda_{ 19 }-\lambda_{ 24 }-\lambda_{ 25 }-\lambda_{ 26 }+\lambda_{
27 }+ 3 \lambda_{ 31 }-\lambda_{ 33 }-\lambda_{ 34 }+\lambda_{ 35 }-3
\lambda_{ 37 }+\lambda_{ 41 }+\lambda_{ 42 }-\lambda_{ 43 }-\lambda_{ 44
}+\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{ 47 }-\lambda_{ 48 }\big]g_{ 31
}+% 31
\big[\lambda_{ 1 }+\lambda_{ 4 }+\lambda_{ 6 }+\lambda_{ 8 }-\lambda_{ 9
}-\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{ 13 }-\lambda_{
14 }+\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17 }+\lambda_{ 18
}-\lambda_{ 20 }-\lambda_{ 22 }-\lambda_{ 25 }+\lambda_{ 26 }-\lambda_{
27 }-\lambda_{ 30 }+ 3 \lambda_{ 32 }-\lambda_{ 34 }-3 \lambda_{ 36
}+\lambda_{ 38 }-\lambda_{ 41 }+\lambda_{ 42 }-\lambda_{ 43 }+\lambda_{
44 }-\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{ 47 }-\lambda_{ 48 }\big]g_{
32 }+% 32
\big[\lambda_{ 1 }+\lambda_{ 2 }+\lambda_{ 7 }-\lambda_{ 8 }+\lambda_{ 9
}-\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }-\lambda_{ 13 }+\lambda_{
14 }+\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17 }+\lambda_{ 18
}-\lambda_{ 19 }-\lambda_{ 24 }-\lambda_{ 25 }-\lambda_{ 26 }+\lambda_{
27 }-\lambda_{ 31 }+ 3 \lambda_{ 33 }-\lambda_{ 34 }-3 \lambda_{ 35
}+\lambda_{ 37 }-\lambda_{ 41 }-\lambda_{ 42 }+\lambda_{ 43 }+\lambda_{
44 }-\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{ 47 }+\lambda_{ 48 }\big]g_{
33 }+% 33
\big[-3 \lambda_{ 1 }-\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4
}-\lambda_{ 5 }-\lambda_{ 6 }-\lambda_{ 7 }-\lambda_{ 16 }-\lambda_{ 17
}-\lambda_{ 18 }+\lambda_{ 19 }+\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{
22 }+\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 25 }+\lambda_{ 26
}+\lambda_{ 27 }-\lambda_{ 28 }-\lambda_{ 29 }-\lambda_{ 30 }-\lambda_{
31 }-\lambda_{ 32 }-\lambda_{ 33 }+ 3 \lambda_{ 34 }+\lambda_{ 35
}+\lambda_{ 36 }+\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39 }+\lambda_{
40 }\big]g_{ 34 }+% 34
\big[-\lambda_{ 1 }-\lambda_{ 2 }-\lambda_{ 7 }+\lambda_{ 8 }-\lambda_{
9 }+\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }+\lambda_{ 13
}-\lambda_{ 14 }-\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17 }-\lambda_{
18 }+\lambda_{ 19 }+\lambda_{ 24 }+\lambda_{ 25 }+\lambda_{ 26
}-\lambda_{ 27 }+\lambda_{ 31 }-3 \lambda_{ 33 }+\lambda_{ 34 }+ 3
\lambda_{ 35 }-\lambda_{ 37 }+\lambda_{ 41 }+\lambda_{ 42 }-\lambda_{ 43
}-\lambda_{ 44 }+\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{ 47 }-\lambda_{
48 }\big]g_{ 35}+% 35
\big[-\lambda_{ 1 }-\lambda_{ 4 }-\lambda_{ 6 }-\lambda_{ 8 }+\lambda_{
9 }+\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{ 13
}+\lambda_{ 14 }-\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17 }-\lambda_{
18 }+\lambda_{ 20 }+\lambda_{ 22 }+\lambda_{ 25 }-\lambda_{ 26
}+\lambda_{ 27 }+\lambda_{ 30 }-3 \lambda_{ 32 }+\lambda_{ 34 }+ 3
\lambda_{ 36 }-\lambda_{ 38 }+\lambda_{ 41 }-\lambda_{ 42 }+\lambda_{ 43
}-\lambda_{ 44 }+\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{ 47 }+\lambda_{
48 }\big]g_{ 36 }+% 36
\big[-\lambda_{ 1 }-\lambda_{ 2 }-\lambda_{ 7 }-\lambda_{ 8 }+\lambda_{
9 }-\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }-\lambda_{ 13
}+\lambda_{ 14 }+\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17 }-\lambda_{
18 }+\lambda_{ 19 }+\lambda_{ 24 }+\lambda_{ 25 }+\lambda_{ 26
}-\lambda_{ 27 }-3 \lambda_{ 31 }+\lambda_{ 33 }+\lambda_{ 34
}-\lambda_{ 35 }+ 3 \lambda_{ 37 }-\lambda_{ 41 }-\lambda_{ 42
}+\lambda_{ 43 }+\lambda_{ 44 }-\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{
47 }+\lambda_{ 48 }\big]g_{ 37}+% 37
\big[-\lambda_{ 1 }-\lambda_{ 4 }-\lambda_{ 6 }+\lambda_{ 8 }-\lambda_{
9 }-\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{ 13
}-\lambda_{ 14 }+\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17 }-\lambda_{
18 }+\lambda_{ 20 }+\lambda_{ 22 }+\lambda_{ 25 }-\lambda_{ 26
}+\lambda_{ 27 }-3 \lambda_{ 30 }+\lambda_{ 32 }+\lambda_{ 34
}-\lambda_{ 36 }+ 3 \lambda_{ 38 }-\lambda_{ 41 }+\lambda_{ 42
}-\lambda_{ 43 }+\lambda_{ 44 }-\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{
47 }-\lambda_{ 48 }\big]g_{ 38 }+% 38
\big[-\lambda_{ 1 }-\lambda_{ 3 }-\lambda_{ 5 }+\lambda_{ 8 }-\lambda_{
9 }-\lambda_{ 10 }+\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{ 13
}+\lambda_{ 14 }-\lambda_{ 15 }-\lambda_{ 16 }-\lambda_{ 17 }+\lambda_{
18 }+\lambda_{ 21 }+\lambda_{ 23 }-\lambda_{ 25 }+\lambda_{ 26
}+\lambda_{ 27 }+\lambda_{ 28 }-3 \lambda_{ 29 }+\lambda_{ 34 }+ 3
\lambda_{ 39 }-\lambda_{ 40 }+\lambda_{ 41 }-\lambda_{ 42 }+\lambda_{ 43
}-\lambda_{ 44 }-\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{ 47 }-\lambda_{
48 }\big]g_{ 39 }+% 39
\big[-\lambda_{ 1 }-\lambda_{ 3 }-\lambda_{ 5 }-\lambda_{ 8 }+\lambda_{
9 }+\lambda_{ 10 }-\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{ 13
}-\lambda_{ 14 }+\lambda_{ 15 }-\lambda_{ 16 }-\lambda_{ 17 }+\lambda_{
18 }+\lambda_{ 21 }+\lambda_{ 23 }-\lambda_{ 25 }+\lambda_{ 26
}+\lambda_{ 27 }-3 \lambda_{ 28 }+\lambda_{ 29 }+\lambda_{ 34
}-\lambda_{ 39 }+ 3 \lambda_{ 40 }-\lambda_{ 41 }+\lambda_{ 42
}-\lambda_{ 43 }+\lambda_{ 44 }+\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{
47 }+\lambda_{ 48 }\big]g_{ 40 }+% 40
\big[\lambda_{ 2 }+\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{ 5 }+\lambda_{ 6
}-\lambda_{ 7 }+\lambda_{ 8 }+\lambda_{ 10 }+\lambda_{ 14 }-3 \lambda_{
15 }+\lambda_{ 19 }-\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{ 22
}-\lambda_{ 23 }-\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29 }+\lambda_{
30 }+\lambda_{ 31 }-\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{ 35
}+\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }+\lambda_{ 39 }-\lambda_{
40 }+ 3 \lambda_{ 41 }-\lambda_{ 42 }-\lambda_{ 46 }-\lambda_{ 48
}\big]g_{ 41 }+% 41
\big[\lambda_{ 2 }-\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{ 5 }-\lambda_{ 6
}-\lambda_{ 7 }+\lambda_{ 8 }+\lambda_{ 10 }-3 \lambda_{ 14 }+\lambda_{
15 }+\lambda_{ 19 }+\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{ 22
}+\lambda_{ 23 }-\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29 }-\lambda_{
30 }+\lambda_{ 31 }+\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{ 35
}-\lambda_{ 36 }-\lambda_{ 37 }+\lambda_{ 38 }-\lambda_{ 39 }+\lambda_{
40 }-\lambda_{ 41 }+ 3 \lambda_{ 42 }-\lambda_{ 46 }-\lambda_{ 48
}\big]g_{ 42 }+% 42
\big[\lambda_{ 2 }-\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{ 5 }-\lambda_{ 6
}-\lambda_{ 7 }+\lambda_{ 9 }+\lambda_{ 11 }+\lambda_{ 12 }-3 \lambda_{
13 }+\lambda_{ 19 }+\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{ 22
}+\lambda_{ 23 }-\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29 }+\lambda_{
30 }-\lambda_{ 31 }-\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{ 35
}+\lambda_{ 36 }+\lambda_{ 37 }-\lambda_{ 38 }+\lambda_{ 39 }-\lambda_{
40 }+ 3 \lambda_{ 43 }-\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 47
}\big]g_{ 43 }+% 43
\big[\lambda_{ 2 }+\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{ 5 }+\lambda_{ 6
}-\lambda_{ 7 }+\lambda_{ 9 }+\lambda_{ 11 }-3 \lambda_{ 12 }+\lambda_{
13 }+\lambda_{ 19 }-\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{ 22
}-\lambda_{ 23 }-\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29 }-\lambda_{
30 }-\lambda_{ 31 }+\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{ 35
}-\lambda_{ 36 }+\lambda_{ 37 }+\lambda_{ 38 }-\lambda_{ 39 }+\lambda_{
40 }-\lambda_{ 43 }+ 3 \lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 47
}\big]g_{ 44 }+% 44
\big[-\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }-\lambda_{ 5 }-\lambda_{
6 }+\lambda_{ 7 }+\lambda_{ 9 }-3 \lambda_{ 11 }+\lambda_{ 12
}+\lambda_{ 13 }-\lambda_{ 19 }+\lambda_{ 20 }+\lambda_{ 21 }-\lambda_{
22 }-\lambda_{ 23 }+\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29
}+\lambda_{ 30 }+\lambda_{ 31 }-\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{
35 }+\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39
}+\lambda_{ 40 }-\lambda_{ 43 }-\lambda_{ 44 }+ 3 \lambda_{ 45
}-\lambda_{ 47 }\big]g_{ 45 }+% 45
\big[-\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }-\lambda_{ 5 }-\lambda_{
6 }+\lambda_{ 7 }+\lambda_{ 8 }-3 \lambda_{ 10 }+\lambda_{ 14
}+\lambda_{ 15 }-\lambda_{ 19 }+\lambda_{ 20 }+\lambda_{ 21 }-\lambda_{
22 }-\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29
}-\lambda_{ 30 }-\lambda_{ 31 }+\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{
35 }-\lambda_{ 36 }+\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39
}-\lambda_{ 40 }-\lambda_{ 41 }-\lambda_{ 42 }+ 3 \lambda_{ 46
}-\lambda_{ 48 }\big]g_{ 46 }+% 46
\big[-\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }+\lambda_{ 5 }+\lambda_{
6 }+\lambda_{ 7 }-3 \lambda_{ 9 }+\lambda_{ 11 }+\lambda_{ 12
}+\lambda_{ 13 }-\lambda_{ 19 }-\lambda_{ 20 }-\lambda_{ 21 }+\lambda_{
22 }+\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29
}-\lambda_{ 30 }+\lambda_{ 31 }+\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{
35 }-\lambda_{ 36 }-\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39
}-\lambda_{ 40 }-\lambda_{ 43 }-\lambda_{ 44 }-\lambda_{ 45 }+ 3
\lambda_{ 47 }\big]g_{ 47 }+% 47
\big[-\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }+\lambda_{ 5 }+\lambda_{
6 }+\lambda_{ 7 }-3 \lambda_{ 8 }+\lambda_{ 10 }+\lambda_{ 14
}+\lambda_{ 15 }-\lambda_{ 19 }-\lambda_{ 20 }-\lambda_{ 21 }+\lambda_{
22 }+\lambda_{ 23 }+\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29
}+\lambda_{ 30 }-\lambda_{ 31 }-\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{
35 }+\lambda_{ 36 }+\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39
}+\lambda_{ 40 }-\lambda_{ 41 }-\lambda_{ 42 }-\lambda_{ 46 }+ 3
\lambda_{ 48 }\big]g_{ 48 }\).
\(uv_{4}=\frac{1}{48}\big[ 3 \lambda_{ 1
}-\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{ 5 }-\lambda_{ 6
}-\lambda_{ 7 }-\lambda_{ 16 }-\lambda_{ 17 }-\lambda_{ 18 }-\lambda_{
19 }-\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{ 22 }-\lambda_{ 23
}-\lambda_{ 24 }-\lambda_{ 25 }-\lambda_{ 26 }-\lambda_{ 27 }+\lambda_{
28 }+\lambda_{ 29 }+\lambda_{ 30 }+\lambda_{ 31 }+\lambda_{ 32
}+\lambda_{ 33 }+ 3 \lambda_{ 34 }+\lambda_{ 35 }+\lambda_{ 36
}+\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39 }+\lambda_{ 40
}\big]\big(g_{ 1 }+g_{ 34 }\big)+% 1
\big[-\lambda_{ 1 }+ 3 \lambda_{ 2 }-\lambda_{ 7 }-\lambda_{ 8
}-\lambda_{ 9 }-\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }+\lambda_{
13 }+\lambda_{ 14 }+\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17
}+\lambda_{ 18 }-\lambda_{ 19 }+ 3 \lambda_{ 24 }+\lambda_{ 25
}+\lambda_{ 26 }-\lambda_{ 27 }-\lambda_{ 31 }-\lambda_{ 33 }-\lambda_{
34 }-\lambda_{ 35 }-\lambda_{ 37 }+\lambda_{ 41 }+\lambda_{ 42
}+\lambda_{ 43 }+\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{
47 }-\lambda_{ 48 }\big]\big(g_{ 2 }+g_{ 24 }\big)+% 2
\big[-\lambda_{ 1 }+ 3 \lambda_{ 3 }-\lambda_{ 5 }-\lambda_{ 8
}-\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{
13 }-\lambda_{ 14 }+\lambda_{ 15 }+\lambda_{ 16 }+\lambda_{ 17
}-\lambda_{ 18 }-\lambda_{ 21 }+ 3 \lambda_{ 23 }-\lambda_{ 25
}+\lambda_{ 26 }+\lambda_{ 27 }-\lambda_{ 28 }-\lambda_{ 29 }-\lambda_{
34 }-\lambda_{ 39 }-\lambda_{ 40 }+\lambda_{ 41 }-\lambda_{ 42
}-\lambda_{ 43 }+\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{
47 }-\lambda_{ 48 }\big]\big(g_{ 3 }+g_{ 23 }\big)+% 3
\big[-\lambda_{ 1 }+ 3 \lambda_{ 4 }-\lambda_{ 6 }-\lambda_{ 8
}-\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{
13 }+\lambda_{ 14 }-\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17
}+\lambda_{ 18 }-\lambda_{ 20 }+ 3 \lambda_{ 22 }+\lambda_{ 25
}-\lambda_{ 26 }+\lambda_{ 27 }-\lambda_{ 30 }-\lambda_{ 32 }-\lambda_{
34 }-\lambda_{ 36 }-\lambda_{ 38 }-\lambda_{ 41 }+\lambda_{ 42
}+\lambda_{ 43 }-\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{
47 }-\lambda_{ 48 }\big]\big(g_{ 4 }+g_{ 22 }\big)+% 4
\big[-\lambda_{ 1 }-\lambda_{ 3 }+ 3 \lambda_{ 5 }+\lambda_{ 8
}+\lambda_{ 9 }-\lambda_{ 10 }-\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{
13 }+\lambda_{ 14 }-\lambda_{ 15 }+\lambda_{ 16 }+\lambda_{ 17
}-\lambda_{ 18 }+ 3 \lambda_{ 21 }-\lambda_{ 23 }-\lambda_{ 25
}+\lambda_{ 26 }+\lambda_{ 27 }-\lambda_{ 28 }-\lambda_{ 29 }-\lambda_{
34 }-\lambda_{ 39 }-\lambda_{ 40 }-\lambda_{ 41 }+\lambda_{ 42
}+\lambda_{ 43 }-\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{
47 }+\lambda_{ 48 }\big]\big(g_{ 5 }+g_{ 21 }\big)+% 5
\big[-\lambda_{ 1 }-\lambda_{ 4 }+ 3 \lambda_{ 6 }+\lambda_{ 8
}+\lambda_{ 9 }-\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{
13 }-\lambda_{ 14 }+\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17
}+\lambda_{ 18 }+ 3 \lambda_{ 20 }-\lambda_{ 22 }+\lambda_{ 25
}-\lambda_{ 26 }+\lambda_{ 27 }-\lambda_{ 30 }-\lambda_{ 32 }-\lambda_{
34 }-\lambda_{ 36 }-\lambda_{ 38 }+\lambda_{ 41 }-\lambda_{ 42
}-\lambda_{ 43 }+\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{
47 }+\lambda_{ 48 }\big]\big(g_{ 6 }+g_{ 20 }\big)+% 6
\big[-\lambda_{ 1 }-\lambda_{ 2 }+ 3 \lambda_{ 7 }+\lambda_{ 8
}+\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }-\lambda_{
13 }-\lambda_{ 14 }-\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17
}+\lambda_{ 18 }+ 3 \lambda_{ 19 }-\lambda_{ 24 }+\lambda_{ 25
}+\lambda_{ 26 }-\lambda_{ 27 }-\lambda_{ 31 }-\lambda_{ 33 }-\lambda_{
34 }-\lambda_{ 35 }-\lambda_{ 37 }-\lambda_{ 41 }-\lambda_{ 42
}-\lambda_{ 43 }-\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{
47 }+\lambda_{ 48 }\big]\big(g_{ 7 }+g_{ 19 }\big)+% 7
\big[-\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }+\lambda_{ 5 }+\lambda_{
6 }+\lambda_{ 7 }+ 3 \lambda_{ 8 }-\lambda_{ 10 }-\lambda_{ 14
}-\lambda_{ 15 }+\lambda_{ 19 }+\lambda_{ 20 }+\lambda_{ 21 }-\lambda_{
22 }-\lambda_{ 23 }-\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29
}-\lambda_{ 30 }+\lambda_{ 31 }+\lambda_{ 32 }-\lambda_{ 33 }-\lambda_{
35 }+\lambda_{ 36 }+\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39
}+\lambda_{ 40 }-\lambda_{ 41 }-\lambda_{ 42 }-\lambda_{ 46 }+ 3
\lambda_{ 48 }\big]\big(g_{ 8 }+g_{ 48 }\big)+% 8
\big[-\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }+\lambda_{ 5 }+\lambda_{
6 }+\lambda_{ 7 }+ 3 \lambda_{ 9 }-\lambda_{ 11 }-\lambda_{ 12
}-\lambda_{ 13 }+\lambda_{ 19 }+\lambda_{ 20 }+\lambda_{ 21 }-\lambda_{
22 }-\lambda_{ 23 }-\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29
}+\lambda_{ 30 }-\lambda_{ 31 }-\lambda_{ 32 }+\lambda_{ 33 }+\lambda_{
35 }-\lambda_{ 36 }-\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39
}-\lambda_{ 40 }-\lambda_{ 43 }-\lambda_{ 44 }-\lambda_{ 45 }+ 3
\lambda_{ 47 }\big]\big(g_{ 9 }+g_{ 47 }\big)+% 9
\big[-\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }-\lambda_{ 5 }-\lambda_{
6 }+\lambda_{ 7 }-\lambda_{ 8 }+ 3 \lambda_{ 10 }-\lambda_{ 14
}-\lambda_{ 15 }+\lambda_{ 19 }-\lambda_{ 20 }-\lambda_{ 21 }+\lambda_{
22 }+\lambda_{ 23 }-\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29
}+\lambda_{ 30 }+\lambda_{ 31 }-\lambda_{ 32 }-\lambda_{ 33 }-\lambda_{
35 }-\lambda_{ 36 }+\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39
}-\lambda_{ 40 }-\lambda_{ 41 }-\lambda_{ 42 }+ 3 \lambda_{ 46
}-\lambda_{ 48 }\big]\big(g_{ 10 }+g_{ 46 }\big)+% 10
\big[-\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }-\lambda_{ 5 }-\lambda_{
6 }+\lambda_{ 7 }-\lambda_{ 9 }+ 3 \lambda_{ 11 }-\lambda_{ 12
}-\lambda_{ 13 }+\lambda_{ 19 }-\lambda_{ 20 }-\lambda_{ 21 }+\lambda_{
22 }+\lambda_{ 23 }-\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29
}-\lambda_{ 30 }-\lambda_{ 31 }+\lambda_{ 32 }+\lambda_{ 33 }+\lambda_{
35 }+\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39
}+\lambda_{ 40 }-\lambda_{ 43 }-\lambda_{ 44 }+ 3 \lambda_{ 45
}-\lambda_{ 47 }\big]\big(g_{ 11 }+g_{ 45 }\big)+% 11
\big[\lambda_{ 2 }+\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{ 5 }+\lambda_{ 6
}-\lambda_{ 7 }-\lambda_{ 9 }-\lambda_{ 11 }+ 3 \lambda_{ 12 }-\lambda_{
13 }-\lambda_{ 19 }+\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{ 22
}+\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29 }+\lambda_{
30 }+\lambda_{ 31 }-\lambda_{ 32 }-\lambda_{ 33 }-\lambda_{ 35
}-\lambda_{ 36 }+\lambda_{ 37 }+\lambda_{ 38 }-\lambda_{ 39 }+\lambda_{
40 }-\lambda_{ 43 }+ 3 \lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 47
}\big]\big(g_{ 12 }+g_{ 44 }\big)+% 12
\big[\lambda_{ 2 }-\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{ 5 }-\lambda_{ 6
}-\lambda_{ 7 }-\lambda_{ 9 }-\lambda_{ 11 }-\lambda_{ 12 }+ 3 \lambda_{
13 }-\lambda_{ 19 }-\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{ 22
}-\lambda_{ 23 }+\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29 }-\lambda_{
30 }+\lambda_{ 31 }+\lambda_{ 32 }-\lambda_{ 33 }-\lambda_{ 35
}+\lambda_{ 36 }+\lambda_{ 37 }-\lambda_{ 38 }+\lambda_{ 39 }-\lambda_{
40 }+ 3 \lambda_{ 43 }-\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 47
}\big]\big(g_{ 13 }+g_{ 43 }\big)+% 13
\big[\lambda_{ 2 }-\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{ 5 }-\lambda_{ 6
}-\lambda_{ 7 }-\lambda_{ 8 }-\lambda_{ 10 }+ 3 \lambda_{ 14 }-\lambda_{
15 }-\lambda_{ 19 }-\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{ 22
}-\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29 }+\lambda_{
30 }-\lambda_{ 31 }-\lambda_{ 32 }+\lambda_{ 33 }+\lambda_{ 35
}-\lambda_{ 36 }-\lambda_{ 37 }+\lambda_{ 38 }-\lambda_{ 39 }+\lambda_{
40 }-\lambda_{ 41 }+ 3 \lambda_{ 42 }-\lambda_{ 46 }-\lambda_{ 48
}\big]\big(g_{ 14 }+g_{ 42 }\big)+% 14
\big[\lambda_{ 2 }+\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{ 5 }+\lambda_{ 6
}-\lambda_{ 7 }-\lambda_{ 8 }-\lambda_{ 10 }-\lambda_{ 14 }+ 3 \lambda_{
15 }-\lambda_{ 19 }+\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{ 22
}+\lambda_{ 23 }+\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29 }-\lambda_{
30 }-\lambda_{ 31 }+\lambda_{ 32 }+\lambda_{ 33 }+\lambda_{ 35
}+\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }+\lambda_{ 39 }-\lambda_{
40 }+ 3 \lambda_{ 41 }-\lambda_{ 42 }-\lambda_{ 46 }-\lambda_{ 48
}\big]\big(g_{ 15 }+g_{ 41 }\big)+% 15
\big[-\lambda_{ 1 }-\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{
5 }+\lambda_{ 6 }-\lambda_{ 7 }+ 3 \lambda_{ 16 }-\lambda_{ 17
}-\lambda_{ 18 }-\lambda_{ 19 }+\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{
22 }+\lambda_{ 23 }-\lambda_{ 24 }-\lambda_{ 25 }-\lambda_{ 26 }+ 3
\lambda_{ 27 }-\lambda_{ 28 }-\lambda_{ 29 }-\lambda_{ 30 }+\lambda_{ 31
}-\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{ 34 }+\lambda_{ 35 }-\lambda_{
36 }+\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39 }-\lambda_{ 40
}\big]\big(g_{ 16 }+g_{ 27 }\big)+% 16
\big[-\lambda_{ 1 }+\lambda_{ 2 }+\lambda_{ 3 }-\lambda_{ 4 }+\lambda_{
5 }-\lambda_{ 6 }+\lambda_{ 7 }-\lambda_{ 16 }+ 3 \lambda_{ 17
}-\lambda_{ 18 }+\lambda_{ 19 }-\lambda_{ 20 }+\lambda_{ 21 }-\lambda_{
22 }+\lambda_{ 23 }+\lambda_{ 24 }-\lambda_{ 25 }+ 3 \lambda_{ 26
}-\lambda_{ 27 }-\lambda_{ 28 }-\lambda_{ 29 }+\lambda_{ 30 }-\lambda_{
31 }+\lambda_{ 32 }-\lambda_{ 33 }-\lambda_{ 34 }-\lambda_{ 35
}+\lambda_{ 36 }-\lambda_{ 37 }+\lambda_{ 38 }-\lambda_{ 39 }-\lambda_{
40 }\big]\big(g_{ 17 }+g_{ 26 }\big)+% 17
\big[-\lambda_{ 1 }+\lambda_{ 2 }-\lambda_{ 3 }+\lambda_{ 4 }-\lambda_{
5 }+\lambda_{ 6 }+\lambda_{ 7 }-\lambda_{ 16 }-\lambda_{ 17 }+ 3
\lambda_{ 18 }+\lambda_{ 19 }+\lambda_{ 20 }-\lambda_{ 21 }+\lambda_{ 22
}-\lambda_{ 23 }+\lambda_{ 24 }+ 3 \lambda_{ 25 }-\lambda_{ 26
}-\lambda_{ 27 }+\lambda_{ 28 }+\lambda_{ 29 }-\lambda_{ 30 }-\lambda_{
31 }-\lambda_{ 32 }-\lambda_{ 33 }-\lambda_{ 34 }-\lambda_{ 35
}-\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }+\lambda_{ 39 }+\lambda_{
40 }\big]\big(g_{ 18 }+g_{ 25 }\big)+% 18
\big[\lambda_{ 1 }-\lambda_{ 3 }-\lambda_{ 5 }+\lambda_{ 8 }-\lambda_{ 9
}-\lambda_{ 10 }+\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{ 13 }+\lambda_{
14 }-\lambda_{ 15 }-\lambda_{ 16 }-\lambda_{ 17 }+\lambda_{ 18
}-\lambda_{ 21 }-\lambda_{ 23 }+\lambda_{ 25 }-\lambda_{ 26 }-\lambda_{
27 }+ 3 \lambda_{ 28 }-\lambda_{ 29 }+\lambda_{ 34 }-\lambda_{ 39 }+ 3
\lambda_{ 40 }-\lambda_{ 41 }+\lambda_{ 42 }-\lambda_{ 43 }+\lambda_{ 44
}+\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{ 47 }+\lambda_{ 48
}\big]\big(g_{ 28 }+g_{ 40 }\big)+% 19
\big[\lambda_{ 1 }-\lambda_{ 3 }-\lambda_{ 5 }-\lambda_{ 8 }+\lambda_{ 9
}+\lambda_{ 10 }-\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{ 13 }-\lambda_{
14 }+\lambda_{ 15 }-\lambda_{ 16 }-\lambda_{ 17 }+\lambda_{ 18
}-\lambda_{ 21 }-\lambda_{ 23 }+\lambda_{ 25 }-\lambda_{ 26 }-\lambda_{
27 }-\lambda_{ 28 }+ 3 \lambda_{ 29 }+\lambda_{ 34 }+ 3 \lambda_{ 39
}-\lambda_{ 40 }+\lambda_{ 41 }-\lambda_{ 42 }+\lambda_{ 43 }-\lambda_{
44 }-\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{ 47 }-\lambda_{ 48
}\big]\big(g_{ 29 }+g_{ 39 }\big)+% 20
\big[\lambda_{ 1 }-\lambda_{ 4 }-\lambda_{ 6 }-\lambda_{ 8 }+\lambda_{ 9
}+\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{ 13 }+\lambda_{
14 }-\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17 }-\lambda_{ 18
}-\lambda_{ 20 }-\lambda_{ 22 }-\lambda_{ 25 }+\lambda_{ 26 }-\lambda_{
27 }+ 3 \lambda_{ 30 }-\lambda_{ 32 }+\lambda_{ 34 }-\lambda_{ 36 }+ 3
\lambda_{ 38 }-\lambda_{ 41 }+\lambda_{ 42 }-\lambda_{ 43 }+\lambda_{ 44
}-\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{ 47 }-\lambda_{ 48
}\big]\big(g_{ 30 }+g_{ 38 }\big)+% 21
\big[\lambda_{ 1 }-\lambda_{ 2 }-\lambda_{ 7 }+\lambda_{ 8 }-\lambda_{ 9
}+\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }+\lambda_{ 13 }-\lambda_{
14 }-\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17 }-\lambda_{ 18
}-\lambda_{ 19 }-\lambda_{ 24 }-\lambda_{ 25 }-\lambda_{ 26 }+\lambda_{
27 }+ 3 \lambda_{ 31 }-\lambda_{ 33 }+\lambda_{ 34 }-\lambda_{ 35 }+ 3
\lambda_{ 37 }-\lambda_{ 41 }-\lambda_{ 42 }+\lambda_{ 43 }+\lambda_{ 44
}-\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{ 47 }+\lambda_{ 48
}\big]\big(g_{ 31 }+g_{ 37 }\big)+% 22
\big[\lambda_{ 1 }-\lambda_{ 4 }-\lambda_{ 6 }+\lambda_{ 8 }-\lambda_{ 9
}-\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{ 13 }-\lambda_{
14 }+\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17 }-\lambda_{ 18
}-\lambda_{ 20 }-\lambda_{ 22 }-\lambda_{ 25 }+\lambda_{ 26 }-\lambda_{
27 }-\lambda_{ 30 }+ 3 \lambda_{ 32 }+\lambda_{ 34 }+ 3 \lambda_{ 36
}-\lambda_{ 38 }+\lambda_{ 41 }-\lambda_{ 42 }+\lambda_{ 43 }-\lambda_{
44 }+\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{ 47 }+\lambda_{ 48
}\big]\big(g_{ 32 }+g_{ 36 }\big)+% 23
\big[\lambda_{ 1 }-\lambda_{ 2 }-\lambda_{ 7 }-\lambda_{ 8 }+\lambda_{ 9
}-\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }-\lambda_{ 13 }+\lambda_{
14 }+\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17 }-\lambda_{ 18
}-\lambda_{ 19 }-\lambda_{ 24 }-\lambda_{ 25 }-\lambda_{ 26 }+\lambda_{
27 }-\lambda_{ 31 }+ 3 \lambda_{ 33 }+\lambda_{ 34 }+ 3 \lambda_{ 35
}-\lambda_{ 37 }+\lambda_{ 41 }+\lambda_{ 42 }-\lambda_{ 43 }-\lambda_{
44 }+\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{ 47 }-\lambda_{ 48
}\big]\big(g_{ 33 }+g_{ 35 }\big)\).
\(uv_{5}=\frac{1}{48}\big[ 3 \lambda_{ 1
}+\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{ 5 }+\lambda_{ 6
}+\lambda_{ 7 }-\lambda_{ 16 }-\lambda_{ 17 }-\lambda_{ 18 }+\lambda_{
19 }+\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{ 22 }+\lambda_{ 23
}+\lambda_{ 24 }-\lambda_{ 25 }-\lambda_{ 26 }-\lambda_{ 27 }-\lambda_{
28 }-\lambda_{ 29 }-\lambda_{ 30 }-\lambda_{ 31 }-\lambda_{ 32
}-\lambda_{ 33 }+ 3 \lambda_{ 34 }-\lambda_{ 35 }-\lambda_{ 36
}-\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39 }-\lambda_{ 40
}\big]\big(g_{ 1 }+g_{ 34 }\big)+% 1
\big[\lambda_{ 1 }+ 3 \lambda_{ 2 }-\lambda_{ 7 }+\lambda_{ 8
}+\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }-\lambda_{
13 }-\lambda_{ 14 }-\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17
}-\lambda_{ 18 }-\lambda_{ 19 }+ 3 \lambda_{ 24 }-\lambda_{ 25
}-\lambda_{ 26 }+\lambda_{ 27 }-\lambda_{ 31 }-\lambda_{ 33 }+\lambda_{
34 }-\lambda_{ 35 }-\lambda_{ 37 }-\lambda_{ 41 }-\lambda_{ 42
}-\lambda_{ 43 }-\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{
47 }+\lambda_{ 48 }\big]\big(g_{ 2 }+g_{ 24 }\big)+% 2
\big[\lambda_{ 1 }+ 3 \lambda_{ 3 }-\lambda_{ 5 }+\lambda_{ 8
}+\lambda_{ 9 }-\lambda_{ 10 }-\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{
13 }+\lambda_{ 14 }-\lambda_{ 15 }-\lambda_{ 16 }-\lambda_{ 17
}+\lambda_{ 18 }-\lambda_{ 21 }+ 3 \lambda_{ 23 }+\lambda_{ 25
}-\lambda_{ 26 }-\lambda_{ 27 }-\lambda_{ 28 }-\lambda_{ 29 }+\lambda_{
34 }-\lambda_{ 39 }-\lambda_{ 40 }-\lambda_{ 41 }+\lambda_{ 42
}+\lambda_{ 43 }-\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{
47 }+\lambda_{ 48 }\big]\big(g_{ 3 }+g_{ 23 }\big)+% 3
\big[\lambda_{ 1 }+ 3 \lambda_{ 4 }-\lambda_{ 6 }+\lambda_{ 8
}+\lambda_{ 9 }-\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{
13 }-\lambda_{ 14 }+\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17
}-\lambda_{ 18 }-\lambda_{ 20 }+ 3 \lambda_{ 22 }-\lambda_{ 25
}+\lambda_{ 26 }-\lambda_{ 27 }-\lambda_{ 30 }-\lambda_{ 32 }+\lambda_{
34 }-\lambda_{ 36 }-\lambda_{ 38 }+\lambda_{ 41 }-\lambda_{ 42
}-\lambda_{ 43 }+\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{
47 }+\lambda_{ 48 }\big]\big(g_{ 4 }+g_{ 22 }\big)+% 4
\big[\lambda_{ 1 }-\lambda_{ 3 }+ 3 \lambda_{ 5 }-\lambda_{ 8
}-\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{
13 }-\lambda_{ 14 }+\lambda_{ 15 }-\lambda_{ 16 }-\lambda_{ 17
}+\lambda_{ 18 }+ 3 \lambda_{ 21 }-\lambda_{ 23 }+\lambda_{ 25
}-\lambda_{ 26 }-\lambda_{ 27 }-\lambda_{ 28 }-\lambda_{ 29 }+\lambda_{
34 }-\lambda_{ 39 }-\lambda_{ 40 }+\lambda_{ 41 }-\lambda_{ 42
}-\lambda_{ 43 }+\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{
47 }-\lambda_{ 48 }\big]\big(g_{ 5 }+g_{ 21 }\big)+% 5
\big[\lambda_{ 1 }-\lambda_{ 4 }+ 3 \lambda_{ 6 }-\lambda_{ 8
}-\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{
13 }+\lambda_{ 14 }-\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17
}-\lambda_{ 18 }+ 3 \lambda_{ 20 }-\lambda_{ 22 }-\lambda_{ 25
}+\lambda_{ 26 }-\lambda_{ 27 }-\lambda_{ 30 }-\lambda_{ 32 }+\lambda_{
34 }-\lambda_{ 36 }-\lambda_{ 38 }-\lambda_{ 41 }+\lambda_{ 42
}+\lambda_{ 43 }-\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{
47 }-\lambda_{ 48 }\big]\big(g_{ 6 }+g_{ 20 }\big)+% 6
\big[\lambda_{ 1 }-\lambda_{ 2 }+ 3 \lambda_{ 7 }-\lambda_{ 8
}-\lambda_{ 9 }-\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }+\lambda_{
13 }+\lambda_{ 14 }+\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17
}-\lambda_{ 18 }+ 3 \lambda_{ 19 }-\lambda_{ 24 }-\lambda_{ 25
}-\lambda_{ 26 }+\lambda_{ 27 }-\lambda_{ 31 }-\lambda_{ 33 }+\lambda_{
34 }-\lambda_{ 35 }-\lambda_{ 37 }+\lambda_{ 41 }+\lambda_{ 42
}+\lambda_{ 43 }+\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{
47 }-\lambda_{ 48 }\big]\big(g_{ 7 }+g_{ 19 }\big)+% 7
\big[\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }-\lambda_{ 5 }-\lambda_{ 6
}-\lambda_{ 7 }+ 3 \lambda_{ 8 }-\lambda_{ 10 }-\lambda_{ 14 }-\lambda_{
15 }-\lambda_{ 19 }-\lambda_{ 20 }-\lambda_{ 21 }+\lambda_{ 22
}+\lambda_{ 23 }+\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29 }+\lambda_{
30 }-\lambda_{ 31 }-\lambda_{ 32 }+\lambda_{ 33 }+\lambda_{ 35
}-\lambda_{ 36 }-\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39 }-\lambda_{
40 }-\lambda_{ 41 }-\lambda_{ 42 }-\lambda_{ 46 }+ 3 \lambda_{ 48
}\big]\big(g_{ 8 }+g_{ 48 }\big)+% 8
\big[\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }-\lambda_{ 5 }-\lambda_{ 6
}-\lambda_{ 7 }+ 3 \lambda_{ 9 }-\lambda_{ 11 }-\lambda_{ 12 }-\lambda_{
13 }-\lambda_{ 19 }-\lambda_{ 20 }-\lambda_{ 21 }+\lambda_{ 22
}+\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29 }-\lambda_{
30 }+\lambda_{ 31 }+\lambda_{ 32 }-\lambda_{ 33 }-\lambda_{ 35
}+\lambda_{ 36 }+\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39 }+\lambda_{
40 }-\lambda_{ 43 }-\lambda_{ 44 }-\lambda_{ 45 }+ 3 \lambda_{ 47
}\big]\big(g_{ 9 }+g_{ 47 }\big)+% 9
\big[\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }+\lambda_{ 5 }+\lambda_{ 6
}-\lambda_{ 7 }-\lambda_{ 8 }+ 3 \lambda_{ 10 }-\lambda_{ 14 }-\lambda_{
15 }-\lambda_{ 19 }+\lambda_{ 20 }+\lambda_{ 21 }-\lambda_{ 22
}-\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29 }-\lambda_{
30 }-\lambda_{ 31 }+\lambda_{ 32 }+\lambda_{ 33 }+\lambda_{ 35
}+\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39 }+\lambda_{
40 }-\lambda_{ 41 }-\lambda_{ 42 }+ 3 \lambda_{ 46 }-\lambda_{ 48
}\big]\big(g_{ 10 }+g_{ 46 }\big)+% 10
\big[\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }+\lambda_{ 5 }+\lambda_{ 6
}-\lambda_{ 7 }-\lambda_{ 9 }+ 3 \lambda_{ 11 }-\lambda_{ 12 }-\lambda_{
13 }-\lambda_{ 19 }+\lambda_{ 20 }+\lambda_{ 21 }-\lambda_{ 22
}-\lambda_{ 23 }+\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29 }+\lambda_{
30 }+\lambda_{ 31 }-\lambda_{ 32 }-\lambda_{ 33 }-\lambda_{ 35
}-\lambda_{ 36 }+\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39 }-\lambda_{
40 }-\lambda_{ 43 }-\lambda_{ 44 }+ 3 \lambda_{ 45 }-\lambda_{ 47
}\big]\big(g_{ 11 }+g_{ 45 }\big)+% 11
\big[-\lambda_{ 2 }-\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{ 5 }-\lambda_{
6 }+\lambda_{ 7 }-\lambda_{ 9 }-\lambda_{ 11 }+ 3 \lambda_{ 12
}-\lambda_{ 13 }+\lambda_{ 19 }-\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{
22 }-\lambda_{ 23 }-\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29
}-\lambda_{ 30 }-\lambda_{ 31 }+\lambda_{ 32 }+\lambda_{ 33 }+\lambda_{
35 }+\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }+\lambda_{ 39
}-\lambda_{ 40 }-\lambda_{ 43 }+ 3 \lambda_{ 44 }-\lambda_{ 45
}-\lambda_{ 47 }\big]\big(g_{ 12 }+g_{ 44 }\big)+% 12
-\lambda_{ 2 }+\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{ 5 }+\lambda_{ 6
}+\lambda_{ 7 }-\lambda_{ 9 }-\lambda_{ 11 }-\lambda_{ 12 }+ 3 \lambda_{
13 }+\lambda_{ 19 }+\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{ 22
}+\lambda_{ 23 }-\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29 }+\lambda_{
30 }-\lambda_{ 31 }-\lambda_{ 32 }+\lambda_{ 33 }+\lambda_{ 35
}-\lambda_{ 36 }-\lambda_{ 37 }+\lambda_{ 38 }-\lambda_{ 39 }+\lambda_{
40 }+ 3 \lambda_{ 43 }-\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 47
}\big]\big(g_{ 13 }+g_{ 43 }\big)+% 13
\big[-\lambda_{ 2 }+\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{ 5 }+\lambda_{
6 }+\lambda_{ 7 }-\lambda_{ 8 }-\lambda_{ 10 }+ 3 \lambda_{ 14
}-\lambda_{ 15 }+\lambda_{ 19 }+\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{
22 }+\lambda_{ 23 }-\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29
}-\lambda_{ 30 }+\lambda_{ 31 }+\lambda_{ 32 }-\lambda_{ 33 }-\lambda_{
35 }+\lambda_{ 36 }+\lambda_{ 37 }-\lambda_{ 38 }+\lambda_{ 39
}-\lambda_{ 40 }-\lambda_{ 41 }+ 3 \lambda_{ 42 }-\lambda_{ 46
}-\lambda_{ 48 }\big]\big(g_{ 14 }+g_{ 42 }\big)+% 14
\big[-\lambda_{ 2 }-\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{ 5 }-\lambda_{
6 }+\lambda_{ 7 }-\lambda_{ 8 }-\lambda_{ 10 }-\lambda_{ 14 }+ 3
\lambda_{ 15 }+\lambda_{ 19 }-\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{ 22
}-\lambda_{ 23 }-\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29 }+\lambda_{
30 }+\lambda_{ 31 }-\lambda_{ 32 }-\lambda_{ 33 }-\lambda_{ 35
}-\lambda_{ 36 }+\lambda_{ 37 }+\lambda_{ 38 }-\lambda_{ 39 }+\lambda_{
40 }+ 3 \lambda_{ 41 }-\lambda_{ 42 }-\lambda_{ 46 }-\lambda_{ 48
}\big]\big(g_{ 15 }+g_{ 41 }\big)+% 15
\big[-\lambda_{ 1 }+\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{
5 }-\lambda_{ 6 }+\lambda_{ 7 }+ 3 \lambda_{ 16 }-\lambda_{ 17
}-\lambda_{ 18 }+\lambda_{ 19 }-\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{
22 }-\lambda_{ 23 }+\lambda_{ 24 }-\lambda_{ 25 }-\lambda_{ 26 }+ 3
\lambda_{ 27 }+\lambda_{ 28 }+\lambda_{ 29 }+\lambda_{ 30 }-\lambda_{ 31
}+\lambda_{ 32 }-\lambda_{ 33 }-\lambda_{ 34 }-\lambda_{ 35 }+\lambda_{
36 }-\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39 }+\lambda_{ 40
}\big]\big(g_{ 16 }+g_{ 27 }\big)+% 16
\big[-\lambda_{ 1 }-\lambda_{ 2 }-\lambda_{ 3 }+\lambda_{ 4 }-\lambda_{
5 }+\lambda_{ 6 }-\lambda_{ 7 }-\lambda_{ 16 }+ 3 \lambda_{ 17
}-\lambda_{ 18 }-\lambda_{ 19 }+\lambda_{ 20 }-\lambda_{ 21 }+\lambda_{
22 }-\lambda_{ 23 }-\lambda_{ 24 }-\lambda_{ 25 }+ 3 \lambda_{ 26
}-\lambda_{ 27 }+\lambda_{ 28 }+\lambda_{ 29 }-\lambda_{ 30 }+\lambda_{
31 }-\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{ 34 }+\lambda_{ 35
}-\lambda_{ 36 }+\lambda_{ 37 }-\lambda_{ 38 }+\lambda_{ 39 }+\lambda_{
40 }\big]\big(g_{ 17 }+g_{ 26 }\big)+% 17
\big[-\lambda_{ 1 }-\lambda_{ 2 }+\lambda_{ 3 }-\lambda_{ 4 }+\lambda_{
5 }-\lambda_{ 6 }-\lambda_{ 7 }-\lambda_{ 16 }-\lambda_{ 17 }+ 3
\lambda_{ 18 }-\lambda_{ 19 }-\lambda_{ 20 }+\lambda_{ 21 }-\lambda_{ 22
}+\lambda_{ 23 }-\lambda_{ 24 }+ 3 \lambda_{ 25 }-\lambda_{ 26
}-\lambda_{ 27 }-\lambda_{ 28 }-\lambda_{ 29 }+\lambda_{ 30 }+\lambda_{
31 }+\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{ 34 }+\lambda_{ 35
}+\lambda_{ 36 }+\lambda_{ 37 }+\lambda_{ 38 }-\lambda_{ 39 }-\lambda_{
40 }\big]\big(g_{ 18 }+g_{ 25 }\big)+% 18
\big[-\lambda_{ 1 }-\lambda_{ 3 }-\lambda_{ 5 }-\lambda_{ 8 }+\lambda_{
9 }+\lambda_{ 10 }-\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{ 13
}-\lambda_{ 14 }+\lambda_{ 15 }+\lambda_{ 16 }+\lambda_{ 17 }-\lambda_{
18 }-\lambda_{ 21 }-\lambda_{ 23 }-\lambda_{ 25 }+\lambda_{ 26
}+\lambda_{ 27 }+ 3 \lambda_{ 28 }-\lambda_{ 29 }-\lambda_{ 34
}-\lambda_{ 39 }+ 3 \lambda_{ 40 }+\lambda_{ 41 }-\lambda_{ 42
}+\lambda_{ 43 }-\lambda_{ 44 }-\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{
47 }-\lambda_{ 48 }\big]\big(g_{ 28 }+g_{ 40 }\big)+% 19
\big[-\lambda_{ 1 }-\lambda_{ 3 }-\lambda_{ 5 }+\lambda_{ 8 }-\lambda_{
9 }-\lambda_{ 10 }+\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{ 13
}+\lambda_{ 14 }-\lambda_{ 15 }+\lambda_{ 16 }+\lambda_{ 17 }-\lambda_{
18 }-\lambda_{ 21 }-\lambda_{ 23 }-\lambda_{ 25 }+\lambda_{ 26
}+\lambda_{ 27 }-\lambda_{ 28 }+ 3 \lambda_{ 29 }-\lambda_{ 34 }+ 3
\lambda_{ 39 }-\lambda_{ 40 }-\lambda_{ 41 }+\lambda_{ 42 }-\lambda_{ 43
}+\lambda_{ 44 }+\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{ 47 }+\lambda_{
48 }\big]\big(g_{ 29 }+g_{ 39 }\big)+% 20
\big[-\lambda_{ 1 }-\lambda_{ 4 }-\lambda_{ 6 }+\lambda_{ 8 }-\lambda_{
9 }-\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{ 13
}-\lambda_{ 14 }+\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17 }+\lambda_{
18 }-\lambda_{ 20 }-\lambda_{ 22 }+\lambda_{ 25 }-\lambda_{ 26
}+\lambda_{ 27 }+ 3 \lambda_{ 30 }-\lambda_{ 32 }-\lambda_{ 34
}-\lambda_{ 36 }+ 3 \lambda_{ 38 }+\lambda_{ 41 }-\lambda_{ 42
}+\lambda_{ 43 }-\lambda_{ 44 }+\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{
47 }+\lambda_{ 48 }\big]\big(g_{ 30 }+g_{ 38 }\big)+% 21
\big[-\lambda_{ 1 }-\lambda_{ 2 }-\lambda_{ 7 }-\lambda_{ 8 }+\lambda_{
9 }-\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }-\lambda_{ 13
}+\lambda_{ 14 }+\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17 }+\lambda_{
18 }-\lambda_{ 19 }-\lambda_{ 24 }+\lambda_{ 25 }+\lambda_{ 26
}-\lambda_{ 27 }+ 3 \lambda_{ 31 }-\lambda_{ 33 }-\lambda_{ 34
}-\lambda_{ 35 }+ 3 \lambda_{ 37 }+\lambda_{ 41 }+\lambda_{ 42
}-\lambda_{ 43 }-\lambda_{ 44 }+\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{
47 }-\lambda_{ 48 }\big]\big(g_{ 31 }+g_{ 37 }\big)+% 22
\big[-\lambda_{ 1 }-\lambda_{ 4 }-\lambda_{ 6 }-\lambda_{ 8 }+\lambda_{
9 }+\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{ 13
}+\lambda_{ 14 }-\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17 }+\lambda_{
18 }-\lambda_{ 20 }-\lambda_{ 22 }+\lambda_{ 25 }-\lambda_{ 26
}+\lambda_{ 27 }-\lambda_{ 30 }+ 3 \lambda_{ 32 }-\lambda_{ 34 }+ 3
\lambda_{ 36 }-\lambda_{ 38 }-\lambda_{ 41 }+\lambda_{ 42 }-\lambda_{ 43
}+\lambda_{ 44 }-\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{ 47 }-\lambda_{
48 }\big]\big(g_{ 32 }+g_{ 36 }\big)+% 23
\big[-\lambda_{ 1 }-\lambda_{ 2 }-\lambda_{ 7 }+\lambda_{ 8 }-\lambda_{
9 }+\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }+\lambda_{ 13
}-\lambda_{ 14 }-\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17 }+\lambda_{
18 }-\lambda_{ 19 }-\lambda_{ 24 }+\lambda_{ 25 }+\lambda_{ 26
}-\lambda_{ 27 }-\lambda_{ 31 }+ 3 \lambda_{ 33 }-\lambda_{ 34 }+ 3
\lambda_{ 35 }-\lambda_{ 37 }-\lambda_{ 41 }-\lambda_{ 42 }+\lambda_{ 43
}+\lambda_{ 44 }-\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{ 47 }+\lambda_{
48 }\big]\big(g_{ 33 }+g_{ 35 }\big)\).
\(uv_{6}=\frac{1}{48}\big[ 3 \lambda_{ 1
}-\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{ 5 }-\lambda_{ 6
}-\lambda_{ 7 }+\lambda_{ 16 }+\lambda_{ 17 }+\lambda_{ 18 }+\lambda_{
19 }+\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{ 22 }+\lambda_{ 23
}+\lambda_{ 24 }-\lambda_{ 25 }-\lambda_{ 26 }-\lambda_{ 27 }-\lambda_{
28 }-\lambda_{ 29 }-\lambda_{ 30 }-\lambda_{ 31 }-\lambda_{ 32
}-\lambda_{ 33 }-3 \lambda_{ 34 }+\lambda_{ 35 }+\lambda_{ 36
}+\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39 }+\lambda_{ 40 }\big]g_{ 1
}+% 1
\big[-\lambda_{ 1 }+ 3 \lambda_{ 2 }-\lambda_{ 7 }-\lambda_{ 8
}-\lambda_{ 9 }-\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }+\lambda_{
13 }+\lambda_{ 14 }+\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17
}-\lambda_{ 18 }+\lambda_{ 19 }-3 \lambda_{ 24 }+\lambda_{ 25
}+\lambda_{ 26 }-\lambda_{ 27 }+\lambda_{ 31 }+\lambda_{ 33 }+\lambda_{
34 }-\lambda_{ 35 }-\lambda_{ 37 }-\lambda_{ 41 }-\lambda_{ 42
}-\lambda_{ 43 }-\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{
47 }+\lambda_{ 48 }\big]g_{ 2 }+% 2
\big[-\lambda_{ 1 }+ 3 \lambda_{ 3 }-\lambda_{ 5 }-\lambda_{ 8
}-\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{
13 }-\lambda_{ 14 }+\lambda_{ 15 }-\lambda_{ 16 }-\lambda_{ 17
}+\lambda_{ 18 }+\lambda_{ 21 }-3 \lambda_{ 23 }-\lambda_{ 25
}+\lambda_{ 26 }+\lambda_{ 27 }+\lambda_{ 28 }+\lambda_{ 29 }+\lambda_{
34 }-\lambda_{ 39 }-\lambda_{ 40 }-\lambda_{ 41 }+\lambda_{ 42
}+\lambda_{ 43 }-\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{
47 }+\lambda_{ 48 }\big]g_{ 3 }+% 3
\big[-\lambda_{ 1 }+ 3 \lambda_{ 4 }-\lambda_{ 6 }-\lambda_{ 8
}-\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{
13 }+\lambda_{ 14 }-\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17
}-\lambda_{ 18 }+\lambda_{ 20 }-3 \lambda_{ 22 }+\lambda_{ 25
}-\lambda_{ 26 }+\lambda_{ 27 }+\lambda_{ 30 }+\lambda_{ 32 }+\lambda_{
34 }-\lambda_{ 36 }-\lambda_{ 38 }+\lambda_{ 41 }-\lambda_{ 42
}-\lambda_{ 43 }+\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{
47 }+\lambda_{ 48 }\big]g_{ 4 }+% 4
\big[-\lambda_{ 1 }-\lambda_{ 3 }+ 3 \lambda_{ 5 }+\lambda_{ 8
}+\lambda_{ 9 }-\lambda_{ 10 }-\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{
13 }+\lambda_{ 14 }-\lambda_{ 15 }-\lambda_{ 16 }-\lambda_{ 17
}+\lambda_{ 18 }-3 \lambda_{ 21 }+\lambda_{ 23 }-\lambda_{ 25
}+\lambda_{ 26 }+\lambda_{ 27 }+\lambda_{ 28 }+\lambda_{ 29 }+\lambda_{
34 }-\lambda_{ 39 }-\lambda_{ 40 }+\lambda_{ 41 }-\lambda_{ 42
}-\lambda_{ 43 }+\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{
47 }-\lambda_{ 48 }\big]g_{ 5 }+% 5
\big[-\lambda_{ 1 }-\lambda_{ 4 }+ 3 \lambda_{ 6 }+\lambda_{ 8
}+\lambda_{ 9 }-\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{
13 }-\lambda_{ 14 }+\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17
}-\lambda_{ 18 }-3 \lambda_{ 20 }+\lambda_{ 22 }+\lambda_{ 25
}-\lambda_{ 26 }+\lambda_{ 27 }+\lambda_{ 30 }+\lambda_{ 32 }+\lambda_{
34 }-\lambda_{ 36 }-\lambda_{ 38 }-\lambda_{ 41 }+\lambda_{ 42
}+\lambda_{ 43 }-\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{
47 }-\lambda_{ 48 }\big]g_{ 6 }+% 6
\big[-\lambda_{ 1 }-\lambda_{ 2 }+ 3 \lambda_{ 7 }+\lambda_{ 8
}+\lambda_{ 9 }+\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }-\lambda_{
13 }-\lambda_{ 14 }-\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17
}-\lambda_{ 18 }-3 \lambda_{ 19 }+\lambda_{ 24 }+\lambda_{ 25
}+\lambda_{ 26 }-\lambda_{ 27 }+\lambda_{ 31 }+\lambda_{ 33 }+\lambda_{
34 }-\lambda_{ 35 }-\lambda_{ 37 }+\lambda_{ 41 }+\lambda_{ 42
}+\lambda_{ 43 }+\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{
47 }-\lambda_{ 48 }\big]g_{ 7 }+% 7
\big[-\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }+\lambda_{ 5 }+\lambda_{
6 }+\lambda_{ 7 }+ 3 \lambda_{ 8 }-\lambda_{ 10 }-\lambda_{ 14
}-\lambda_{ 15 }-\lambda_{ 19 }-\lambda_{ 20 }-\lambda_{ 21 }+\lambda_{
22 }+\lambda_{ 23 }+\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29
}+\lambda_{ 30 }-\lambda_{ 31 }-\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{
35 }+\lambda_{ 36 }+\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39
}+\lambda_{ 40 }+\lambda_{ 41 }+\lambda_{ 42 }+\lambda_{ 46 }-3
\lambda_{ 48 }\big]g_{ 8 }+% 8
\big[-\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }+\lambda_{ 5 }+\lambda_{
6 }+\lambda_{ 7 }+ 3 \lambda_{ 9 }-\lambda_{ 11 }-\lambda_{ 12
}-\lambda_{ 13 }-\lambda_{ 19 }-\lambda_{ 20 }-\lambda_{ 21 }+\lambda_{
22 }+\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29
}-\lambda_{ 30 }+\lambda_{ 31 }+\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{
35 }-\lambda_{ 36 }-\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39
}-\lambda_{ 40 }+\lambda_{ 43 }+\lambda_{ 44 }+\lambda_{ 45 }-3
\lambda_{ 47 }\big]g_{ 9 }+% 9
\big[-\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }-\lambda_{ 5 }-\lambda_{
6 }+\lambda_{ 7 }-\lambda_{ 8 }+ 3 \lambda_{ 10 }-\lambda_{ 14
}-\lambda_{ 15 }-\lambda_{ 19 }+\lambda_{ 20 }+\lambda_{ 21 }-\lambda_{
22 }-\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29
}-\lambda_{ 30 }-\lambda_{ 31 }+\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{
35 }-\lambda_{ 36 }+\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39
}-\lambda_{ 40 }+\lambda_{ 41 }+\lambda_{ 42 }-3 \lambda_{ 46
}+\lambda_{ 48 }\big]g_{ 10 }+% 10
\big[-\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }-\lambda_{ 5 }-\lambda_{
6 }+\lambda_{ 7 }-\lambda_{ 9 }+ 3 \lambda_{ 11 }-\lambda_{ 12
}-\lambda_{ 13 }-\lambda_{ 19 }+\lambda_{ 20 }+\lambda_{ 21 }-\lambda_{
22 }-\lambda_{ 23 }+\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29
}+\lambda_{ 30 }+\lambda_{ 31 }-\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{
35 }+\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39
}+\lambda_{ 40 }+\lambda_{ 43 }+\lambda_{ 44 }-3 \lambda_{ 45
}+\lambda_{ 47 }\big]g_{ 11 }+% 11
\big[\lambda_{ 2 }+\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{ 5 }+\lambda_{ 6
}-\lambda_{ 7 }-\lambda_{ 9 }-\lambda_{ 11 }+ 3 \lambda_{ 12 }-\lambda_{
13 }+\lambda_{ 19 }-\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{ 22
}-\lambda_{ 23 }-\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29 }-\lambda_{
30 }-\lambda_{ 31 }+\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{ 35
}-\lambda_{ 36 }+\lambda_{ 37 }+\lambda_{ 38 }-\lambda_{ 39 }+\lambda_{
40 }+\lambda_{ 43 }-3 \lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 47
}\big]g_{ 12 }+% 12
\big[\lambda_{ 2 }-\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{ 5 }-\lambda_{ 6
}-\lambda_{ 7 }-\lambda_{ 9 }-\lambda_{ 11 }-\lambda_{ 12 }+ 3 \lambda_{
13 }+\lambda_{ 19 }+\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{ 22
}+\lambda_{ 23 }-\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29 }+\lambda_{
30 }-\lambda_{ 31 }-\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{ 35
}+\lambda_{ 36 }+\lambda_{ 37 }-\lambda_{ 38 }+\lambda_{ 39 }-\lambda_{
40 }-3 \lambda_{ 43 }+\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 47
}\big]g_{ 13 }+% 13
\big[\lambda_{ 2 }-\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{ 5 }-\lambda_{ 6
}-\lambda_{ 7 }-\lambda_{ 8 }-\lambda_{ 10 }+ 3 \lambda_{ 14 }-\lambda_{
15 }+\lambda_{ 19 }+\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{ 22
}+\lambda_{ 23 }-\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29 }-\lambda_{
30 }+\lambda_{ 31 }+\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{ 35
}-\lambda_{ 36 }-\lambda_{ 37 }+\lambda_{ 38 }-\lambda_{ 39 }+\lambda_{
40 }+\lambda_{ 41 }-3 \lambda_{ 42 }+\lambda_{ 46 }+\lambda_{ 48
}\big]g_{ 14 }+% 14
\big[\lambda_{ 2 }+\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{ 5 }+\lambda_{ 6
}-\lambda_{ 7 }-\lambda_{ 8 }-\lambda_{ 10 }-\lambda_{ 14 }+ 3 \lambda_{
15 }+\lambda_{ 19 }-\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{ 22
}-\lambda_{ 23 }-\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29 }+\lambda_{
30 }+\lambda_{ 31 }-\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{ 35
}+\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }+\lambda_{ 39 }-\lambda_{
40 }-3 \lambda_{ 41 }+\lambda_{ 42 }+\lambda_{ 46 }+\lambda_{ 48
}\big]g_{ 15 }+% 15
\big[\lambda_{ 1 }+\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{ 5
}-\lambda_{ 6 }+\lambda_{ 7 }+ 3 \lambda_{ 16 }-\lambda_{ 17 }-\lambda_{
18 }-\lambda_{ 19 }+\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{ 22
}+\lambda_{ 23 }-\lambda_{ 24 }+\lambda_{ 25 }+\lambda_{ 26 }-3
\lambda_{ 27 }-\lambda_{ 28 }-\lambda_{ 29 }-\lambda_{ 30 }+\lambda_{ 31
}-\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{ 34 }-\lambda_{ 35 }+\lambda_{
36 }-\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39 }+\lambda_{ 40 }\big]g_{
16 }+% 16
\big[\lambda_{ 1 }-\lambda_{ 2 }-\lambda_{ 3 }+\lambda_{ 4 }-\lambda_{ 5
}+\lambda_{ 6 }-\lambda_{ 7 }-\lambda_{ 16 }+ 3 \lambda_{ 17 }-\lambda_{
18 }+\lambda_{ 19 }-\lambda_{ 20 }+\lambda_{ 21 }-\lambda_{ 22
}+\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 25 }-3 \lambda_{ 26
}+\lambda_{ 27 }-\lambda_{ 28 }-\lambda_{ 29 }+\lambda_{ 30 }-\lambda_{
31 }+\lambda_{ 32 }-\lambda_{ 33 }-\lambda_{ 34 }+\lambda_{ 35
}-\lambda_{ 36 }+\lambda_{ 37 }-\lambda_{ 38 }+\lambda_{ 39 }+\lambda_{
40 }\big]g_{ 17 }+% 17
+\lambda_{ 1 }-\lambda_{ 2 }+\lambda_{ 3 }-\lambda_{ 4 }+\lambda_{ 5
}-\lambda_{ 6 }-\lambda_{ 7 }-\lambda_{ 16 }-\lambda_{ 17 }+ 3 \lambda_{
18 }+\lambda_{ 19 }+\lambda_{ 20 }-\lambda_{ 21 }+\lambda_{ 22
}-\lambda_{ 23 }+\lambda_{ 24 }-3 \lambda_{ 25 }+\lambda_{ 26
}+\lambda_{ 27 }+\lambda_{ 28 }+\lambda_{ 29 }-\lambda_{ 30 }-\lambda_{
31 }-\lambda_{ 32 }-\lambda_{ 33 }-\lambda_{ 34 }+\lambda_{ 35
}+\lambda_{ 36 }+\lambda_{ 37 }+\lambda_{ 38 }-\lambda_{ 39 }-\lambda_{
40 }\big]g_{ 18 }+% 18
\big[\lambda_{ 1 }+\lambda_{ 2 }-3 \lambda_{ 7 }-\lambda_{ 8 }-\lambda_{
9 }-\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }+\lambda_{ 13
}+\lambda_{ 14 }+\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17 }+\lambda_{
18 }+ 3 \lambda_{ 19 }-\lambda_{ 24 }-\lambda_{ 25 }-\lambda_{ 26
}+\lambda_{ 27 }-\lambda_{ 31 }-\lambda_{ 33 }-\lambda_{ 34 }+\lambda_{
35 }+\lambda_{ 37 }-\lambda_{ 41 }-\lambda_{ 42 }-\lambda_{ 43
}-\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{ 47 }+\lambda_{
48 }\big]g_{ 19 }+% 19
\big[\lambda_{ 1 }+\lambda_{ 4 }-3 \lambda_{ 6 }-\lambda_{ 8 }-\lambda_{
9 }+\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{ 13
}+\lambda_{ 14 }-\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17 }+\lambda_{
18 }+ 3 \lambda_{ 20 }-\lambda_{ 22 }-\lambda_{ 25 }+\lambda_{ 26
}-\lambda_{ 27 }-\lambda_{ 30 }-\lambda_{ 32 }-\lambda_{ 34 }+\lambda_{
36 }+\lambda_{ 38 }+\lambda_{ 41 }-\lambda_{ 42 }-\lambda_{ 43
}+\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{ 47 }+\lambda_{
48 }\big]g_{ 20 }+% 20
\big[\lambda_{ 1 }+\lambda_{ 3 }-3 \lambda_{ 5 }-\lambda_{ 8 }-\lambda_{
9 }+\lambda_{ 10 }+\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{ 13
}-\lambda_{ 14 }+\lambda_{ 15 }+\lambda_{ 16 }+\lambda_{ 17 }-\lambda_{
18 }+ 3 \lambda_{ 21 }-\lambda_{ 23 }+\lambda_{ 25 }-\lambda_{ 26
}-\lambda_{ 27 }-\lambda_{ 28 }-\lambda_{ 29 }-\lambda_{ 34 }+\lambda_{
39 }+\lambda_{ 40 }-\lambda_{ 41 }+\lambda_{ 42 }+\lambda_{ 43
}-\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{ 47 }+\lambda_{
48 }\big]g_{ 21 }+% 21
\big[\lambda_{ 1 }-3 \lambda_{ 4 }+\lambda_{ 6 }+\lambda_{ 8 }+\lambda_{
9 }-\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{ 13
}-\lambda_{ 14 }+\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17 }+\lambda_{
18 }-\lambda_{ 20 }+ 3 \lambda_{ 22 }-\lambda_{ 25 }+\lambda_{ 26
}-\lambda_{ 27 }-\lambda_{ 30 }-\lambda_{ 32 }-\lambda_{ 34 }+\lambda_{
36 }+\lambda_{ 38 }-\lambda_{ 41 }+\lambda_{ 42 }+\lambda_{ 43
}-\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{ 47 }-\lambda_{
48 }\big]g_{ 22 }+% 22
\big[\lambda_{ 1 }-3 \lambda_{ 3 }+\lambda_{ 5 }+\lambda_{ 8 }+\lambda_{
9 }-\lambda_{ 10 }-\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{ 13
}+\lambda_{ 14 }-\lambda_{ 15 }+\lambda_{ 16 }+\lambda_{ 17 }-\lambda_{
18 }-\lambda_{ 21 }+ 3 \lambda_{ 23 }+\lambda_{ 25 }-\lambda_{ 26
}-\lambda_{ 27 }-\lambda_{ 28 }-\lambda_{ 29 }-\lambda_{ 34 }+\lambda_{
39 }+\lambda_{ 40 }+\lambda_{ 41 }-\lambda_{ 42 }-\lambda_{ 43
}+\lambda_{ 44 }+\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{ 47 }-\lambda_{
48 }\big]g_{ 23 }+% 23
\big[\lambda_{ 1 }-3 \lambda_{ 2 }+\lambda_{ 7 }+\lambda_{ 8 }+\lambda_{
9 }+\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }-\lambda_{ 13
}-\lambda_{ 14 }-\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17 }+\lambda_{
18 }-\lambda_{ 19 }+ 3 \lambda_{ 24 }-\lambda_{ 25 }-\lambda_{ 26
}+\lambda_{ 27 }-\lambda_{ 31 }-\lambda_{ 33 }-\lambda_{ 34 }+\lambda_{
35 }+\lambda_{ 37 }+\lambda_{ 41 }+\lambda_{ 42 }+\lambda_{ 43
}+\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{ 47 }-\lambda_{
48 }\big]g_{ 24 }+% 24
\big[-\lambda_{ 1 }+\lambda_{ 2 }-\lambda_{ 3 }+\lambda_{ 4 }-\lambda_{
5 }+\lambda_{ 6 }+\lambda_{ 7 }+\lambda_{ 16 }+\lambda_{ 17 }-3
\lambda_{ 18 }-\lambda_{ 19 }-\lambda_{ 20 }+\lambda_{ 21 }-\lambda_{ 22
}+\lambda_{ 23 }-\lambda_{ 24 }+ 3 \lambda_{ 25 }-\lambda_{ 26
}-\lambda_{ 27 }-\lambda_{ 28 }-\lambda_{ 29 }+\lambda_{ 30 }+\lambda_{
31 }+\lambda_{ 32 }+\lambda_{ 33 }+\lambda_{ 34 }-\lambda_{ 35
}-\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }+\lambda_{ 39 }+\lambda_{
40 }\big]g_{ 25 }+% 25
\big[-\lambda_{ 1 }+\lambda_{ 2 }+\lambda_{ 3 }-\lambda_{ 4 }+\lambda_{
5 }-\lambda_{ 6 }+\lambda_{ 7 }+\lambda_{ 16 }-3 \lambda_{ 17
}+\lambda_{ 18 }-\lambda_{ 19 }+\lambda_{ 20 }-\lambda_{ 21 }+\lambda_{
22 }-\lambda_{ 23 }-\lambda_{ 24 }-\lambda_{ 25 }+ 3 \lambda_{ 26
}-\lambda_{ 27 }+\lambda_{ 28 }+\lambda_{ 29 }-\lambda_{ 30 }+\lambda_{
31 }-\lambda_{ 32 }+\lambda_{ 33 }+\lambda_{ 34 }-\lambda_{ 35
}+\lambda_{ 36 }-\lambda_{ 37 }+\lambda_{ 38 }-\lambda_{ 39 }-\lambda_{
40 }\big]g_{ 26 }+% 26
\big[-\lambda_{ 1 }-\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{
5 }+\lambda_{ 6 }-\lambda_{ 7 }-3 \lambda_{ 16 }+\lambda_{ 17
}+\lambda_{ 18 }+\lambda_{ 19 }-\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{
22 }-\lambda_{ 23 }+\lambda_{ 24 }-\lambda_{ 25 }-\lambda_{ 26 }+ 3
\lambda_{ 27 }+\lambda_{ 28 }+\lambda_{ 29 }+\lambda_{ 30 }-\lambda_{ 31
}+\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{ 34 }+\lambda_{ 35 }-\lambda_{
36 }+\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39 }-\lambda_{ 40 }\big]g_{
27 }+% 27
\big[-\lambda_{ 1 }+\lambda_{ 3 }+\lambda_{ 5 }-\lambda_{ 8 }+\lambda_{
9 }+\lambda_{ 10 }-\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{ 13
}-\lambda_{ 14 }+\lambda_{ 15 }-\lambda_{ 16 }-\lambda_{ 17 }+\lambda_{
18 }-\lambda_{ 21 }-\lambda_{ 23 }-\lambda_{ 25 }+\lambda_{ 26
}+\lambda_{ 27 }+ 3 \lambda_{ 28 }-\lambda_{ 29 }+\lambda_{ 34
}+\lambda_{ 39 }-3 \lambda_{ 40 }-\lambda_{ 41 }+\lambda_{ 42
}-\lambda_{ 43 }+\lambda_{ 44 }+\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{
47 }+\lambda_{ 48 }\big]g_{ 28 }+% 28
\big[-\lambda_{ 1 }+\lambda_{ 3 }+\lambda_{ 5 }+\lambda_{ 8 }-\lambda_{
9 }-\lambda_{ 10 }+\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{ 13
}+\lambda_{ 14 }-\lambda_{ 15 }-\lambda_{ 16 }-\lambda_{ 17 }+\lambda_{
18 }-\lambda_{ 21 }-\lambda_{ 23 }-\lambda_{ 25 }+\lambda_{ 26
}+\lambda_{ 27 }-\lambda_{ 28 }+ 3 \lambda_{ 29 }+\lambda_{ 34 }-3
\lambda_{ 39 }+\lambda_{ 40 }+\lambda_{ 41 }-\lambda_{ 42 }+\lambda_{ 43
}-\lambda_{ 44 }-\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{ 47 }-\lambda_{
48 }\big]g_{ 29 }+% 29
\big[-\lambda_{ 1 }+\lambda_{ 4 }+\lambda_{ 6 }+\lambda_{ 8 }-\lambda_{
9 }-\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{ 13
}-\lambda_{ 14 }+\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17 }-\lambda_{
18 }-\lambda_{ 20 }-\lambda_{ 22 }+\lambda_{ 25 }-\lambda_{ 26
}+\lambda_{ 27 }+ 3 \lambda_{ 30 }-\lambda_{ 32 }+\lambda_{ 34
}+\lambda_{ 36 }-3 \lambda_{ 38 }-\lambda_{ 41 }+\lambda_{ 42
}-\lambda_{ 43 }+\lambda_{ 44 }-\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{
47 }-\lambda_{ 48 }\big]g_{ 30 }+% 30
\big[-\lambda_{ 1 }+\lambda_{ 2 }+\lambda_{ 7 }-\lambda_{ 8 }+\lambda_{
9 }-\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }-\lambda_{ 13
}+\lambda_{ 14 }+\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17 }-\lambda_{
18 }-\lambda_{ 19 }-\lambda_{ 24 }+\lambda_{ 25 }+\lambda_{ 26
}-\lambda_{ 27 }+ 3 \lambda_{ 31 }-\lambda_{ 33 }+\lambda_{ 34
}+\lambda_{ 35 }-3 \lambda_{ 37 }-\lambda_{ 41 }-\lambda_{ 42
}+\lambda_{ 43 }+\lambda_{ 44 }-\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{
47 }+\lambda_{ 48 }\big]g_{ 31 }+% 31
\big[-\lambda_{ 1 }+\lambda_{ 4 }+\lambda_{ 6 }-\lambda_{ 8 }+\lambda_{
9 }+\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{ 13
}+\lambda_{ 14 }-\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17 }-\lambda_{
18 }-\lambda_{ 20 }-\lambda_{ 22 }+\lambda_{ 25 }-\lambda_{ 26
}+\lambda_{ 27 }-\lambda_{ 30 }+ 3 \lambda_{ 32 }+\lambda_{ 34 }-3
\lambda_{ 36 }+\lambda_{ 38 }+\lambda_{ 41 }-\lambda_{ 42 }+\lambda_{ 43
}-\lambda_{ 44 }+\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{ 47 }+\lambda_{
48 }\big]g_{ 32 }+% 32
\big[-\lambda_{ 1 }+\lambda_{ 2 }+\lambda_{ 7 }+\lambda_{ 8 }-\lambda_{
9 }+\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }+\lambda_{ 13
}-\lambda_{ 14 }-\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17 }-\lambda_{
18 }-\lambda_{ 19 }-\lambda_{ 24 }+\lambda_{ 25 }+\lambda_{ 26
}-\lambda_{ 27 }-\lambda_{ 31 }+ 3 \lambda_{ 33 }+\lambda_{ 34 }-3
\lambda_{ 35 }+\lambda_{ 37 }+\lambda_{ 41 }+\lambda_{ 42 }-\lambda_{ 43
}-\lambda_{ 44 }+\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{ 47 }-\lambda_{
48 }\big]g_{ 33}+% 33
\big[-3 \lambda_{ 1 }+\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4
}+\lambda_{ 5 }+\lambda_{ 6 }+\lambda_{ 7 }-\lambda_{ 16 }-\lambda_{ 17
}-\lambda_{ 18 }-\lambda_{ 19 }-\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{
22 }-\lambda_{ 23 }-\lambda_{ 24 }+\lambda_{ 25 }+\lambda_{ 26
}+\lambda_{ 27 }+\lambda_{ 28 }+\lambda_{ 29 }+\lambda_{ 30 }+\lambda_{
31 }+\lambda_{ 32 }+\lambda_{ 33 }+ 3 \lambda_{ 34 }-\lambda_{ 35
}-\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39 }-\lambda_{
40 }\big]g_{ 34 }+% 34
\big[\lambda_{ 1 }-\lambda_{ 2 }-\lambda_{ 7 }-\lambda_{ 8 }+\lambda_{ 9
}-\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }-\lambda_{ 13 }+\lambda_{
14 }+\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17 }+\lambda_{ 18
}+\lambda_{ 19 }+\lambda_{ 24 }-\lambda_{ 25 }-\lambda_{ 26 }+\lambda_{
27 }+\lambda_{ 31 }-3 \lambda_{ 33 }-\lambda_{ 34 }+ 3 \lambda_{ 35
}-\lambda_{ 37 }-\lambda_{ 41 }-\lambda_{ 42 }+\lambda_{ 43 }+\lambda_{
44 }-\lambda_{ 45 }+\lambda_{ 46 }-\lambda_{ 47 }+\lambda_{ 48 }\big]g_{
35 }+% 35
\big[\lambda_{ 1 }-\lambda_{ 4 }-\lambda_{ 6 }+\lambda_{ 8 }-\lambda_{ 9
}-\lambda_{ 10 }+\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{ 13 }-\lambda_{
14 }+\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17 }+\lambda_{ 18
}+\lambda_{ 20 }+\lambda_{ 22 }-\lambda_{ 25 }+\lambda_{ 26 }-\lambda_{
27 }+\lambda_{ 30 }-3 \lambda_{ 32 }-\lambda_{ 34 }+ 3 \lambda_{ 36
}-\lambda_{ 38 }-\lambda_{ 41 }+\lambda_{ 42 }-\lambda_{ 43 }+\lambda_{
44 }-\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{ 47 }-\lambda_{ 48 }\big]g_{
36 }+% 36
\big[\lambda_{ 1 }-\lambda_{ 2 }-\lambda_{ 7 }+\lambda_{ 8 }-\lambda_{ 9
}+\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }+\lambda_{ 13 }-\lambda_{
14 }-\lambda_{ 15 }-\lambda_{ 16 }+\lambda_{ 17 }+\lambda_{ 18
}+\lambda_{ 19 }+\lambda_{ 24 }-\lambda_{ 25 }-\lambda_{ 26 }+\lambda_{
27 }-3 \lambda_{ 31 }+\lambda_{ 33 }-\lambda_{ 34 }-\lambda_{ 35 }+ 3
\lambda_{ 37 }+\lambda_{ 41 }+\lambda_{ 42 }-\lambda_{ 43 }-\lambda_{ 44
}+\lambda_{ 45 }-\lambda_{ 46 }+\lambda_{ 47 }-\lambda_{ 48 }\big]g_{ 37
}+% 37
\big[\lambda_{ 1 }-\lambda_{ 4 }-\lambda_{ 6 }-\lambda_{ 8 }+\lambda_{ 9
}+\lambda_{ 10 }-\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{ 13 }+\lambda_{
14 }-\lambda_{ 15 }+\lambda_{ 16 }-\lambda_{ 17 }+\lambda_{ 18
}+\lambda_{ 20 }+\lambda_{ 22 }-\lambda_{ 25 }+\lambda_{ 26 }-\lambda_{
27 }-3 \lambda_{ 30 }+\lambda_{ 32 }-\lambda_{ 34 }-\lambda_{ 36 }+ 3
\lambda_{ 38 }+\lambda_{ 41 }-\lambda_{ 42 }+\lambda_{ 43 }-\lambda_{ 44
}+\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{ 47 }+\lambda_{ 48 }\big]g_{ 38
}+% 38
\big[\lambda_{ 1 }-\lambda_{ 3 }-\lambda_{ 5 }-\lambda_{ 8 }+\lambda_{ 9
}+\lambda_{ 10 }-\lambda_{ 11 }-\lambda_{ 12 }+\lambda_{ 13 }-\lambda_{
14 }+\lambda_{ 15 }+\lambda_{ 16 }+\lambda_{ 17 }-\lambda_{ 18
}+\lambda_{ 21 }+\lambda_{ 23 }+\lambda_{ 25 }-\lambda_{ 26 }-\lambda_{
27 }+\lambda_{ 28 }-3 \lambda_{ 29 }-\lambda_{ 34 }+ 3 \lambda_{ 39
}-\lambda_{ 40 }-\lambda_{ 41 }+\lambda_{ 42 }-\lambda_{ 43 }+\lambda_{
44 }+\lambda_{ 45 }-\lambda_{ 46 }-\lambda_{ 47 }+\lambda_{ 48 }\big]g_{
39 }+% 39
\big[\lambda_{ 1 }-\lambda_{ 3 }-\lambda_{ 5 }+\lambda_{ 8 }-\lambda_{ 9
}-\lambda_{ 10 }+\lambda_{ 11 }+\lambda_{ 12 }-\lambda_{ 13 }+\lambda_{
14 }-\lambda_{ 15 }+\lambda_{ 16 }+\lambda_{ 17 }-\lambda_{ 18
}+\lambda_{ 21 }+\lambda_{ 23 }+\lambda_{ 25 }-\lambda_{ 26 }-\lambda_{
27 }-3 \lambda_{ 28 }+\lambda_{ 29 }-\lambda_{ 34 }-\lambda_{ 39 }+ 3
\lambda_{ 40 }+\lambda_{ 41 }-\lambda_{ 42 }+\lambda_{ 43 }-\lambda_{ 44
}-\lambda_{ 45 }+\lambda_{ 46 }+\lambda_{ 47 }-\lambda_{ 48 }\big]g_{ 40
}+% 40
\big[-\lambda_{ 2 }-\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{ 5 }-\lambda_{
6 }+\lambda_{ 7 }+\lambda_{ 8 }+\lambda_{ 10 }+\lambda_{ 14 }-3
\lambda_{ 15 }-\lambda_{ 19 }+\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{ 22
}+\lambda_{ 23 }+\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29 }-\lambda_{
30 }-\lambda_{ 31 }+\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{ 35
}-\lambda_{ 36 }+\lambda_{ 37 }+\lambda_{ 38 }-\lambda_{ 39 }+\lambda_{
40 }+ 3 \lambda_{ 41 }-\lambda_{ 42 }-\lambda_{ 46 }-\lambda_{ 48
}\big]g_{ 41 }+% 41
\big[-\lambda_{ 2 }+\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{ 5 }+\lambda_{
6 }+\lambda_{ 7 }+\lambda_{ 8 }+\lambda_{ 10 }-3 \lambda_{ 14
}+\lambda_{ 15 }-\lambda_{ 19 }-\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{
22 }-\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29
}+\lambda_{ 30 }-\lambda_{ 31 }-\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{
35 }+\lambda_{ 36 }+\lambda_{ 37 }-\lambda_{ 38 }+\lambda_{ 39
}-\lambda_{ 40 }-\lambda_{ 41 }+ 3 \lambda_{ 42 }-\lambda_{ 46
}-\lambda_{ 48 }\big]g_{ 42 }+% 42
\big[-\lambda_{ 2 }+\lambda_{ 3 }-\lambda_{ 4 }-\lambda_{ 5 }+\lambda_{
6 }+\lambda_{ 7 }+\lambda_{ 9 }+\lambda_{ 11 }+\lambda_{ 12 }-3
\lambda_{ 13 }-\lambda_{ 19 }-\lambda_{ 20 }+\lambda_{ 21 }+\lambda_{ 22
}-\lambda_{ 23 }+\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29 }-\lambda_{
30 }+\lambda_{ 31 }+\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{ 35
}-\lambda_{ 36 }-\lambda_{ 37 }+\lambda_{ 38 }-\lambda_{ 39 }+\lambda_{
40 }+ 3 \lambda_{ 43 }-\lambda_{ 44 }-\lambda_{ 45 }-\lambda_{ 47
}\big]g_{ 43 }+% 43
\big[-\lambda_{ 2 }-\lambda_{ 3 }+\lambda_{ 4 }+\lambda_{ 5 }-\lambda_{
6 }+\lambda_{ 7 }+\lambda_{ 9 }+\lambda_{ 11 }-3 \lambda_{ 12
}+\lambda_{ 13 }-\lambda_{ 19 }+\lambda_{ 20 }-\lambda_{ 21 }-\lambda_{
22 }+\lambda_{ 23 }+\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29
}+\lambda_{ 30 }+\lambda_{ 31 }-\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{
35 }+\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }+\lambda_{ 39
}-\lambda_{ 40 }-\lambda_{ 43 }+ 3 \lambda_{ 44 }-\lambda_{ 45
}-\lambda_{ 47 }\big]g_{ 44 }+% 44
\big[\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }+\lambda_{ 5 }+\lambda_{ 6
}-\lambda_{ 7 }+\lambda_{ 9 }-3 \lambda_{ 11 }+\lambda_{ 12 }+\lambda_{
13 }+\lambda_{ 19 }-\lambda_{ 20 }-\lambda_{ 21 }+\lambda_{ 22
}+\lambda_{ 23 }-\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29 }-\lambda_{
30 }-\lambda_{ 31 }+\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{ 35
}-\lambda_{ 36 }+\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39 }-\lambda_{
40 }-\lambda_{ 43 }-\lambda_{ 44 }+ 3 \lambda_{ 45 }-\lambda_{ 47
}\big]g_{ 45 }+% 45
\big[\lambda_{ 2 }-\lambda_{ 3 }-\lambda_{ 4 }+\lambda_{ 5 }+\lambda_{ 6
}-\lambda_{ 7 }+\lambda_{ 8 }-3 \lambda_{ 10 }+\lambda_{ 14 }+\lambda_{
15 }+\lambda_{ 19 }-\lambda_{ 20 }-\lambda_{ 21 }+\lambda_{ 22
}+\lambda_{ 23 }-\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29 }+\lambda_{
30 }+\lambda_{ 31 }-\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{ 35
}+\lambda_{ 36 }-\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39 }+\lambda_{
40 }-\lambda_{ 41 }-\lambda_{ 42 }+ 3 \lambda_{ 46 }-\lambda_{ 48
}\big]g_{ 46 }+% 46
\big[\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }-\lambda_{ 5 }-\lambda_{ 6
}-\lambda_{ 7 }-3 \lambda_{ 9 }+\lambda_{ 11 }+\lambda_{ 12 }+\lambda_{
13 }+\lambda_{ 19 }+\lambda_{ 20 }+\lambda_{ 21 }-\lambda_{ 22
}-\lambda_{ 23 }-\lambda_{ 24 }-\lambda_{ 28 }+\lambda_{ 29 }+\lambda_{
30 }-\lambda_{ 31 }-\lambda_{ 32 }+\lambda_{ 33 }-\lambda_{ 35
}+\lambda_{ 36 }+\lambda_{ 37 }-\lambda_{ 38 }-\lambda_{ 39 }+\lambda_{
40 }-\lambda_{ 43 }-\lambda_{ 44 }-\lambda_{ 45 }+ 3 \lambda_{ 47
}\big]g_{ 47 }+% 47
\big[\lambda_{ 2 }+\lambda_{ 3 }+\lambda_{ 4 }-\lambda_{ 5 }-\lambda_{ 6
}-\lambda_{ 7 }-3 \lambda_{ 8 }+\lambda_{ 10 }+\lambda_{ 14 }+\lambda_{
15 }+\lambda_{ 19 }+\lambda_{ 20 }+\lambda_{ 21 }-\lambda_{ 22
}-\lambda_{ 23 }-\lambda_{ 24 }+\lambda_{ 28 }-\lambda_{ 29 }-\lambda_{
30 }+\lambda_{ 31 }+\lambda_{ 32 }-\lambda_{ 33 }+\lambda_{ 35
}-\lambda_{ 36 }-\lambda_{ 37 }+\lambda_{ 38 }+\lambda_{ 39 }-\lambda_{
40 }-\lambda_{ 41 }-\lambda_{ 42 }-\lambda_{ 46 }+ 3 \lambda_{ 48
}\big]g_{ 48 }\).