Some methods of constructions of square tactical decomposable regular group divisible designs are described. These designs are useful in threshold schemes. An L_2 design is also identified as square tactical decomposable. This completes spectrum of the solutions of entire L_2 designs listed in Clatworthy [2] using matrix approaches.
A block design \(D(v,b,r,k)\) is an arrangement of a set \(X\) of \(v\) elements into \(b\) subsets (blocks) each of size \(k\) such that each element of \(X\) appears \(r\) times. The parameters satisfy the relation \(bk=vr.\)
Let a (0, 1) – matrix \(N\) have a decomposition \[N={\left[N_{ij}\right]}_{ \begin{array}{c} i=1,2,\dots ,s \\ j=1,2,\dots ,t \end{array} }\] where \(N_{ij}\) are submatrices of \(N\) of suitable sizes. The decomposition is called row tactical if row sum of \(N_{ij}\) is \(r_{ij}\) and column tactical if the column sum of \(N_{ij}\) is \(k_{ij}\) and tactical if it is row as well as column tactical. If \(N\) is the incidence matrix of a block design \(D\), then \(D\) is called row (column) tactical decomposable design. \(D\) is called uniform row (column) tactical decomposable if \(r_{ij}=\alpha \left(k_{ij}=\beta \right),\ \)for all \(i,j.\) Further if each \(N_{ij}\) is an \(m\times m\) matrix, \(D\) is called square tactical decomposable design, STD \((m)\) [1, 10].
Interested readers can find several recent constructions of tactical decomposable designs in [8, 9, 7, 10, 5].
We now introduce the notation used in this paper. \(I_n\) denotes the identity matrix of order \(n\), and \(0_{m\times n}\) the null matrix of order \(m\times n\). The juxtaposition \(\bigl[ A \mid B \bigr]\) represents the matrix obtained by placing matrices \(A\) and \(B\) side by side. The vector \(e_n\) is the \(n\times 1\) column vector with all entries equal to \(1\), and \(J_{v\times b}\) is the \(v\times b\) matrix with all entries equal to \(1\) (in particular, \(J_{v\times v} = J_v\)). For a matrix \(N\), we write \(N^T\) for its transpose.
The matrix \(\alpha = \mathrm{circ}(0\ 1\ 0\ 0\ldots 0)\) is a permutation circulant matrix of order \(n\) such that \(\alpha^{n} = I_n\). A square block matrix is called **block–circulant** if each block is a circulant matrix of smaller order. The \(RX\) and \(LSX\) numbers are taken from Clatworthy [2].
Further if \(A=\left(a_{ij}\right)\ \)is an \(m\times n\) matrix and \(\ B=\left(b_{ij}\right)\) is a \(p\times q\) matrix, then their Kronecker product \(A\otimes B\) is the \(mp\times nq\) block matrix: \[A\otimes B=\left( \begin{array}{ccc} a_{11}B & \cdots & a_{1n}B \\ \vdots & \ddots & \vdots \\ a_{m1}B & \cdots & a_{mn}B \end{array} \right).\]
Let \(v=mn\) elements be arranged in an \(m\times n\) array. A group divisible (GD) design is an arrangement of the \(v=mn\) elements in \(b\) blocks each of size \(k\) such that:
1. Every element occurs at most once in a block;
2. Every element occurs in \(r\) blocks;
3. Every pair of elements, which are in the same row of the \(m\times n\) array, occur together in \({\lambda }_1\) blocks whereas remaining pair of elements occur together in \({\lambda }_2\) blocks.
If \(r-{\lambda }_1>0,\ rk-v{\lambda }_2>0,\) the GD design is regular (R). Further let \(N\) be \(v\times b\) incidence matrix of a block design such that \(J_vN=kJ_{v\times b}\) and satisfies the following conditions (i) or (ii):
(i)\[\label{GrindEQ__1_} NN^T=\left(r-{\lambda }_1\right)\left(I_m\mathrm{\otimes }I_n\right)+\left({\lambda }_1-{\lambda }_2\right)\left(I_m\mathrm{\otimes }J_n\right)+{\lambda }_2\left(J_m\mathrm{\otimes }J_n\right). \tag{1}\]
Let \(R_i\ \)and \(R_j\) be any two rows of blocks of \(N\). Then from (1), their inner product is \[R_i\bullet R_j=\left\{ \begin{array}{c} rI_n+{\lambda }_1{\left(J-I\right)}_n;i=j, \\ {\lambda }_2J_n;i\neq j ,\end{array} \right.=\left\{ \begin{array}{c} (r-{\lambda }_1)I_n+{\lambda }_1J_n;i=j, \\ {\lambda }_2J_n;i\neq j. \end{array} \right.\]
(ii)\[\label{GrindEQ__2_} NN^T=\left(r-{\lambda }_2\right)\left(I_n\mathrm{\otimes }I_m\right)+{\lambda }_2\left(J_n\mathrm{\otimes }J_m\right)+\left({\lambda }_1-{\lambda }_2\right)\left\{\left(J_n\mathrm{-}I_n\right)\mathrm{\otimes }I_m\right\}. \tag{2}\]
Then (2) implies, \[R_i\bullet R_j=\left\{ \begin{array}{c} rI_m+{\lambda }_2{\left(J-I\right)}_m;i=j ,\\ {\lambda }_1I_m+{\lambda }_2{\left(J-I\right)}_m;i\neq j ,\end{array} \right.=\left\{ \begin{array}{c} (r-{\lambda }_2)I_m+{\lambda }_2J_m;i=j ,\\ {(\lambda }_1-{\lambda }_2)I_m+{\lambda }_2J_m;i\neq j .\end{array} \right.\]
Then \(N\) represents a GD design with parameters: \(v=mn,r,k,b,{\lambda }_1,{\lambda }_2,m,n\). For GD association schemes, we refer to Clatworthy [5]. A GD design is called STD\((n)\) or STD\((m)\) with orthogonal rows if its incidence matrix satisfies the conditions (i) or (ii) respectively. In some cases it may happen that \(m=n.\)
Example 1.1. Consider a RGD design with parameters: \(v=8,r=10,k=2,b=40,{\lambda }_1=2,{\lambda }_2=1,m=2,n=4\), which is a STD\((n=4)\), with a pair of orthogonal rows. This design is listed as R32 in Clatworthy [2] whose incidence matrix \(N\) is given below. \[N=\left( \begin{array}{cc} M & 0_{4\times 6} \\ 0_{4\times 6} & M \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} M & 0_{4\times 6} \\ 0_{4\times 6} & M \end{array} \begin{array}{cc} M & 0_{4\times 6} \\ 0_{4\times 6} & M \end{array} \begin{array}{c} I_4 \\ I_4 \end{array} \begin{array}{ccc} I_4 & I_4 & I_4 \\ \alpha & {\alpha }^2 & {\alpha }^3 \end{array} }\right.} \begin{array}{cc} M & 0_{4\times 6} \\ 0_{4\times 6} & M \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} M & 0_{4\times 6} \\ 0_{4\times 6} & M \end{array} \begin{array}{cc} M & 0_{4\times 6} \\ 0_{4\times 6} & M \end{array} \begin{array}{c} I_4 \\ I_4 \end{array} \begin{array}{ccc} I_4 & I_4 & I_4 \\ \alpha & {\alpha }^2 & {\alpha }^3 \end{array} }\right.} \begin{array}{c} I_4 \\ I_4 \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} M & 0_{4\times 6} \\ 0_{4\times 6} & M \end{array} \begin{array}{cc} M & 0_{4\times 6} \\ 0_{4\times 6} & M \end{array} \begin{array}{c} I_4 \\ I_4 \end{array} \begin{array}{ccc} I_4 & I_4 & I_4 \\ \alpha & {\alpha }^2 & {\alpha }^3 \end{array} }\right.} \begin{array}{ccc} I_4 & I_4 & I_4 \\ \alpha & {\alpha }^2 & {\alpha }^3 \end{array} \right),\] where \(M\) represents a balanced incomplete block design with parameters: \(v'=4,b'=6,r'=3,k'=2,\lambda =1\) and \(\alpha =circ\ (0\ 1\ 0\ 0)\) is a permutation circulant matrix of order four [see Clatworthy [5]].
Example 1.2. Consider a RGD design with parameters\(:v=22,r=10,k=5,b=44,{\lambda }_1=0,{\lambda }_2=2,m=11,n=2\), which is a STD\((m=11)\), with a pair of orthogonal rows. This design is listed in Freeman [3] whose incidence matrix \(N\) is given below. \[N=\left( \begin{array}{cccc} \alpha +{\alpha }^2+{\alpha }^3+{\alpha }^6 & {\alpha }^9 & \alpha +{\alpha }^3+{\alpha }^8 & {\alpha }^2+{\alpha }^{10} \\ {\alpha }^9 & \alpha +{\alpha }^2+{\alpha }^3+{\alpha }^6 & {\alpha }^2+{\alpha }^{10} & \alpha +{\alpha }^3+{\alpha }^8 \end{array} \right),\] where \(\alpha =circ\left(0\ 1\ 0\ 0\dots 0\right)\) is permutation circulant matrix of order 11.
Let \(v=n^2\)elements be arranged in an \(n\times n\) array. An \(L_2\) design is an arrangement of the \(v=n^2\) elements in \(b\) blocks each of size \(k\) such that:
1. Every element occurs at most once in a block;
2. Every element occurs in \(r\) blocks;
3. Every pair of elements, which are in the same row or in the same column of the \(n\times n\) array, occur together in \({\lambda }_1\) blocks whereas every other pair of elements occur together in \({\lambda }_2\) blocks.
The non–negative integers \(v=n^2,b,r,k,{\lambda }_1\)and \({\lambda }_2\) are known as parameters of the \(L_2\) design and they satisfy the relations: \(bk=vr;2\left(n-1\right){\lambda }_1+{(n-1)}^2{\lambda }_2=r\left(k-1\right).\)
A block design \(D(v,b,r,k)\) whose b blocks can be divided into \(t={r}/{\mu }\) classes, each of size \(\beta ={v\mu }/{k}\) and such that in each class of \(\beta\) blocks every element of D is replicated \(\mu\) times, is called an \(\mu -\) resolvable design. If \(\mu =1\) then the design is said to be resolvable.
Alternatively, if the incidence matrix \(N\) of a block design \(D(v,b,r,k)\) may be partitioned in to submatrices as: \(N=\left(N_1\mathrel{\left|\vphantom{N_1 N_2 \cdots N_t}\right.}N_2\mathrel{\left|\vphantom{N_1 N_2 \cdots N_t}\right.}\cdots \mathrel{\left|\vphantom{N_1 N_2 \cdots N_t}\right.}N_t\right)\) where each \(N_i(1\le i\le t)\) is a \(v\times ({v\mu }/{k})\) matrix such that each row sum of \(N_i\) is \(\mu\) then the design is \(\mu -\)resolvable.
A \(P-\)matrix \(P_{i_1i_2\dots .i_n}\) introduced by Saurabh and Prasad [7], is an \(n\times n\) matrix whose first row is the \(i_1-\mathrm{th}\) row of \(I_n\), second row is the \(i_2-\mathrm{th}\) row of \(I_n\) and so on up to \(n-\)th row where \(i_1,i_2,\dots ,i_n\in A=\left\{1,2,\dots ,n\right\},\) where the symbols \(i_1,i_2,\dots ,i_n\) may or may not be distinct.
Further if all \(i'_ks(1\le k\le n)\) are distinct then \(P_{i_1i_2\dots .i_n}\) is a permutation as well as an orthogonal matrix and hence \(P_{i_1i_2\dots .i_n}{P_{i_1i_2\dots .i_n}}^T=I_n.\)
Example 1.3. \(P_{3134}=\left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right).\) Clearly first row of \(P_{3134}\) is third row of \(I_4\), second row is first row of \(I_4\), third and fourth rows are same as of \(I_4.\)
Example 1.4. \(P_{1423}{P_{1423}}^T=I_4.\) \[P_{1214}{P_{1314}}^T\ =\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right){\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)}^T=\left( \begin{array}{cccc} 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right).\]
Here, some series of STD regular group divisible designs are obtained. An \(L_2\) design is also identified as tactical decomposable. This completes spectrum of the solutions of entire \(L_2\) designs listed in Clatworthy [2] using matrix approaches. Such designs are useful as \(\left(2,\ n\right)-\)threshold schemes [see Saurabh [6]].
Let \(N={\left(N_{ij}\right)}_{ \begin{array}{c} 1\le i\le s \\ 1\le j\le t \end{array} }\) be a block matrix where each \(N_{ij}\) is a \(P-\)matrix. Then \(N\) is the incidence matrix of a GD design if its rows of blocks satisfy either condition (i) or (ii) of Subsection 1.2.
The construction methods of RGD designs using \(P-\)matrices and permutation circulant matrices are described below:
Theorem 2.1. There exist resolvable STD(5) RGD designs with parameters: \[ \left(a\right)\qquad v=10,r=8s+t,k=2,b=5(8s+t),{\lambda }_1=t,{\lambda }_2=s,m=5,n=2, \label{GrindEQ__3_} \ \tag{3}\] \[\left(b\right)\qquad v=15,r=6s+t,k=3,b=5(4s+t),{\lambda }_1=t,{\lambda }_2=s,m=5,n=3,\label{GrindEQ__4_} \tag{4}\] where \(s\ge 1,t\ge 0.\)
Proof. (a) Consider the following block matrix: \[M_1=\left( \begin{array}{cccccccc} P_{11234} & P_{12133} & P_{12314} & P_{12341} & P_{12334} & P_{12234} & P_{12323} & P_{12342} \\ P_{53425} & P_{24545} & P_{34255} & P_{45523} & P_{41552} & P_{55143} & P_{45415} & P_{53451} \end{array} \right).\]
Then we have \(M_1M^T_1=7\left(I_2\mathrm{\otimes }I_5\right)+\left(J_2\mathrm{\otimes }J_5\right)-\left\{\left(J_2\mathrm{-}I_2\right)\mathrm{\otimes }I_5\right\}\) and each column sum of \(M_1\) is 2. Hence \(M_1\) represents a RGD design R36 with parameters: \(v=10,b=40,r=8,k=2,\lambda_1=0,\lambda_2=1,m=5,n=2.\)
Let \(N_1\) be a block matrix obtained by taking \(s\) copies of \(M_1\) i. e. (\(\underbrace{M_1\ M_1\dots M_1}_{s\ times}\)) and \(N_2\) be the block matrix obtained by taking \(t\) copies of the block matrix: \(e_2\otimes I_5=\left( \begin{array}{c} I_5 \\ I_5 \end{array} \right)\), arranged columnwise. Then \(N=\left[N_1\mathrel{\left|\vphantom{N_1 N_2}\right.}N_2\right]\) represents a RGD design with generalized parameters (3) which may be verified. Further since each row sum of the columns of blocks of \(N\) is one, the design is resolvable.
(b) Consider the following block matrix: \[M_1=\left( \begin{array}{cccccc} P_{11133} & P_{13525} & P_{12451} & P_{14314} & P_{14245} & P_{14552} \\ P_{22244} & P_{21453} & P_{23535} & P_{25425} & P_{25112} & P_{25341} \\ P_{34555} & P_{34241} & P_{34124} & P_{31532} & P_{33354} & P_{32413} \end{array} \right).\]
Then we have \(M_1M^T_1=5\left(I_3\mathrm{\otimes }I_5\right)+\left(J_3\mathrm{\otimes }J_5\right)-\left\{\left(J_3\mathrm{-}I_3\right)\mathrm{\otimes }I_5\right\}\) and each column sum of \(M_1\) is 3. Hence \(M_1\) represents RGD design R81 with parameters: \(v=15,b=30,r=6,k=3,\lambda_1=0,\lambda_2=1,m=5,n=3.\)
Let \(N_1\) be a block matrix obtained by taking \(s\) copies of \(M_1\) and \(N_2\) be the block matrix obtained by taking \(t\) copies of the block matrix: \(e_3\otimes I_5=\left( \begin{array}{c} I_5 \\ I_5 \\ I_5 \end{array} \right)\), arranged columnwise. Then \(N=\left[N_1\mathrel{\left|\vphantom{N_1 N_2}\right.}N_2\right]\) represents a RGD design with generalized parameters (4) which may be verified. Further since each row sum of the columns of blocks of \(N\) is one, the design is resolvable.
A Table 1 of tactical decomposable RGD designs obtained using above results is given below:
| No. | \(RX:(v,r,k,b,\lambda_1,\lambda_2,m,n)\) | \((s,t)\) |
|---|---|---|
| 1 | \(R37:(10,\ 10,\ 2,\ 50,\ 2,\ 1,\ 5,\ 2)\) | \((1,\ 2)\) |
| 2 | \(R82:(15,\ 8,\ 3,\ 40,\ 2,\ 1,\ 5,\ 3)\) | \((1,\ 2)\) |
| 3 | \(R84:(15,\ 9,\ 3,\ 45,\ 3,\ 1,\ 5,\ 3)\) | \((1,\ 3)\) |
| 4 | \(R85:(15,\ 10,\ 3,\ 50,\ 4,\ 1,\ 5,\ 3)\) | \((1,\ 4)\) |
 ◻
Theorem 2.2. There exists a \(\left(n+1\right)-\)resolvable STD(5) RGD design with parameters: \[\label{GrindEQ__5_} v=5n,\ r=2\left(n+1\right),k=n+1,b=10n,\lambda_1=2n,\lambda_2=1,m=5,n\ \left(>1\right). \tag{5}\]
Proof. Let \(\alpha=circ\ (0\ 1\ 0\ 0\ 0)\) be a permutation circulant matrix of order 5. Consider the following block matrix: \[N=\left(N_1\mathrel{\left|\vphantom{N_1 N_2}\right.}N_2\right)=\left(I_n\otimes \left(\alpha+\alpha^2\right)+{(J-I)}_n\otimes \alpha\mathrel{\left|\vphantom{I_n\otimes \left(\alpha+\alpha^2\right)+{(J-I)}_n\otimes \alpha I_n\otimes \left(\alpha+\alpha^3\right)+{(J-I)}_n\otimes \alpha}\right.}I_n\otimes \left(\alpha+\alpha^3\right)+{(J-I)}_n\otimes \alpha\right).\]
Then we have \(NN^T=(2n+1)\left(I_n\mathrm{\otimes }I_5\right)+\left(J_n\mathrm{\otimes }J_5\right)+\left(2n-1\right)\left\{\left(J_n\mathrm{-}I_n\right)\mathrm{\otimes }I_5\right\}\) and each column sum of \(N\) is \(n+1\). Hence \(N\) represents a RGD design with parameters (5). Further since each row sum of \(N_1\) and \(N_2\) is \(\left(n+1\right)\), the design is \(\left(n+1\right)-\)resolvable. ◻
Remark 2.3. The series (3) may also be found in Raghavarao and Padgett [4]. Here the plug–in matrices are \(\alpha,\alpha+\alpha^2\) and \(\alpha+\alpha^3\) whereas the plug–in matrices used in Raghavarao and Padgett (2005) are \(0_n,I_n\) and \(J_n\). For \(t=2,\ 3\) and \(4\) in Theorem 2.2, we obtain RGD designs R69 (3–resolvable), R115 (4–resolvable) and R152 (5–resolvable) respectively.
Some STD \((m)\) RGD designs are displayed below where \(\alpha=circ\ \left(0\ 1\ 0\dots 0\right)\ \)is a permutation circulant matrix of order \(m\) and \(N\) represents the incidence matrix.
1. \(R50:v=6,r=9,k=3,b=18,\lambda_1=6,\lambda_2=3,m=3,n=2\) (3\(-\)resolvable). \[N=\left( \begin{array}{cc} I_3 & I_3+\alpha \\ I_3+\alpha & I_3 \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} I_3 & I_3+\alpha \\ I_3+\alpha & I_3 \end{array} \begin{array}{cc} I_3 & I_3+\alpha^2 \\ I_3+\alpha^2 & I_3 \end{array} \begin{array}{cc} I_3 & I_3+\alpha \\ I_3+\alpha & I_3 \end{array} }\right.} \begin{array}{cc} I_3 & I_3+\alpha^2 \\ I_3+\alpha^2 & I_3 \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} I_3 & I_3+\alpha \\ I_3+\alpha & I_3 \end{array} \begin{array}{cc} I_3 & I_3+\alpha^2 \\ I_3+\alpha^2 & I_3 \end{array} \begin{array}{cc} I_3 & I_3+\alpha \\ I_3+\alpha & I_3 \end{array} }\right.} \begin{array}{cc} I_3 & I_3+\alpha \\ I_3+\alpha & I_3 \end{array} \right).\]
2. \(R56:v=8,r=9,k=3,b=24,\lambda_1=6,\lambda_2=2,m=4,n=2\) (3\(-\)resolvable). \[N=\left( \begin{array}{cc} I_4 & I_4+\alpha \\ I_4+\alpha & I_4 \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} I_4 & I_4+\alpha \\ I_4+\alpha & I_4 \end{array} \begin{array}{cc} I_4 & I_4+\alpha^2 \\ I_4+\alpha^2 & I_4 \end{array} \begin{array}{cc} \alpha & \alpha+\alpha^2 \\ \alpha+\alpha^2 & \alpha \end{array} }\right.} \begin{array}{cc} I_4 & I_4+\alpha^2 \\ I_4+\alpha^2 & I_4 \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} I_4 & I_4+\alpha \\ I_4+\alpha & I_4 \end{array} \begin{array}{cc} I_4 & I_4+\alpha^2 \\ I_4+\alpha^2 & I_4 \end{array} \begin{array}{cc} \alpha & \alpha+\alpha^2 \\ \alpha+\alpha^2 & \alpha \end{array} }\right.} \begin{array}{cc} \alpha & \alpha+\alpha^2 \\ \alpha+\alpha^2 & \alpha \end{array} \right).\]
3. \(R80:v=14,r=9,k=3,b=42,\lambda_1=6,\lambda_2=1,m=7,n=2\) (3\(-\)resolvable). \[N=\left( \begin{array}{cc} \alpha+\alpha^2 & \alpha \\ \alpha & \alpha+\alpha^2 \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} \alpha+\alpha^2 & \alpha \\ \alpha & \alpha+\alpha^2 \end{array} \begin{array}{cc} \alpha+\alpha^3 & \alpha \\ \alpha & \alpha+\alpha^3 \end{array} \begin{array}{cc} \alpha+\alpha^4 & \alpha \\ \alpha & \alpha+\alpha^4 \end{array} }\right.} \begin{array}{cc} \alpha+\alpha^3 & \alpha \\ \alpha & \alpha+\alpha^3 \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} \alpha+\alpha^2 & \alpha \\ \alpha & \alpha+\alpha^2 \end{array} \begin{array}{cc} \alpha+\alpha^3 & \alpha \\ \alpha & \alpha+\alpha^3 \end{array} \begin{array}{cc} \alpha+\alpha^4 & \alpha \\ \alpha & \alpha+\alpha^4 \end{array} }\right.} \begin{array}{cc} \alpha+\alpha^4 & \alpha \\ \alpha & \alpha+\alpha^4 \end{array} \right).\]
4. \(R89:v=18,r=9,k=3,b=54,\lambda_1=2,\lambda_2=1,m=9,n=2.\) \[N=\left( \begin{array}{cc} \alpha & \alpha+\alpha^3 \\ \alpha^7+\alpha^8 & \alpha^3 \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} \alpha & \alpha+\alpha^3 \\ \alpha^7+\alpha^8 & \alpha^3 \end{array} \begin{array}{cccc} \alpha & \alpha & \alpha & \alpha+\alpha^2+\alpha^5 \\ \alpha^2+\alpha^4 & \alpha+\alpha^5 & \alpha^6+I_9 & 0_9 \end{array} }\right.} \begin{array}{cccc} \alpha & \alpha & \alpha & \alpha+\alpha^2+\alpha^5 \\ \alpha^2+\alpha^4 & \alpha+\alpha^5 & \alpha^6+I_9 & 0_9 \end{array} \right).\]
5. \(R96:v=6,r=8,k=4,b=12,\lambda_1=4,\lambda_2=5,m=3,n=2\) (4\(-\)resolvable). \[N=\left( \begin{array}{cc} I_3 & J_3 \\ J_3 & I_3 \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} I_3 & J_3 \\ J_3 & I_3 \end{array} \begin{array}{cc} I_3+\alpha & I_3+\alpha^2 \\ I_3+\alpha^2 & I_3+\alpha \end{array} }\right.} \begin{array}{cc} I_3+\alpha & I_3+\alpha^2 \\ I_3+\alpha^2 & I_3+\alpha \end{array} \right).\]
6. \(R117:v=15,r=8,k=4,b=30,\lambda_1=1,\lambda_2=2,m=3,n=5\) (4\(-\)resolvable). \[N=\left[circ\ (\alpha+\alpha^4,\ I_5,I_5)\mathrel{\left|\vphantom{circ\ (\alpha+\alpha^4,\ I_5,I_5) circ\ (\alpha^2+\alpha^3,\ I_5,I_5)}\right.}circ\ (\alpha^2+\alpha^3,\ I_5,I_5)\right],\ \]where \(\alpha=circ\ \left(0\ 1\ 0\ 0\ 0\right).\)
Clearly \(circ\ (\alpha+\alpha^4,\ I_5,I_5)\) and \(circ\ (\alpha^2+\alpha^3,\ I_5,I_5)\) are block–circulant matrices of order 15.
Saurabh and Sinha [8] obtained some series of tactical decomposable \(L_2\) designs which yielded solutions of all designs listed in Clatworthy [2] except LS4: \(v=16,\ r=9,\ k=2,\ b=72,\ \ n_1=6,n_2=9,\lambda_1=0,\lambda_2=1\). The aim was to plug\(-\)in suitable \(n\times t\) matrices in certain combinatorial structures so that the resultant matrix becomes incidence matrix of the desired \(L_2\) design. Here LS4 is identified as a tactical decomposable design whose solution is obtained using a permutation circulant matrix of order four.
Let \(\alpha=circ\ \left(0\ 1\ 0\ 0\right)\ \)and \(0\) be a null matrix of order four. Consider the following \(\left(0,\ 1\right)-\)block matrix \(N=\left(N_1\mathrel{\left|\vphantom{N_1 N_2 N_3 N_4 N_5 N_6 N_7 N_8 N_9}\right.}N_2\mathrel{\left|\vphantom{N_1 N_2 N_3 N_4 N_5 N_6 N_7 N_8 N_9}\right.}N_3\mathrel{\left|\vphantom{N_1 N_2 N_3 N_4 N_5 N_6 N_7 N_8 N_9}\right.}N_4\mathrel{\left|\vphantom{N_1 N_2 N_3 N_4 N_5 N_6 N_7 N_8 N_9}\right.}N_5\mathrel{\left|\vphantom{N_1 N_2 N_3 N_4 N_5 N_6 N_7 N_8 N_9}\right.}N_6\mathrel{\left|\vphantom{N_1 N_2 N_3 N_4 N_5 N_6 N_7 N_8 N_9}\right.}N_7\mathrel{\left|\vphantom{N_1 N_2 N_3 N_4 N_5 N_6 N_7 N_8 N_9}\right.}N_8\mathrel{\left|\vphantom{N_1 N_2 N_3 N_4 N_5 N_6 N_7 N_8 N_9}\right.}N_9\right),\) where: \[\begin{aligned} &N_1=\left( \begin{array}{cc} \alpha & 0 \\ \alpha^2 & 0 \\ 0 & \alpha^3 \\ 0 & I_4 \end{array} \right), N_2=\left( \begin{array}{cc} \alpha^2 & 0 \\ \alpha & 0 \\ 0 & I_4 \\ 0 & \alpha^3 \end{array} \right), N_3=\left( \begin{array}{cc} \alpha & 0 \\ 0 & I_4 \\ \alpha^2 & 0 \\ 0 & \alpha^3 \end{array} \right),\\ &N_4=\left( \begin{array}{cc} \alpha & 0 \\ 0 & \alpha^3 \\ 0 & \alpha \\ \alpha^2 & 0 \end{array} \right), N_5=\left( \begin{array}{cc} 0 & \alpha \\ \alpha^3 & 0 \\ 0 & I_4 \\ I_4 & 0 \end{array} \right), N_6=\left( \begin{array}{cc} \alpha & 0 \\ \alpha^3 & 0 \\ 0 & I_4 \\ 0 & \alpha^2 \end{array} \right),\\ & N_7=\left( \begin{array}{cc} \alpha & 0 \\ 0 & \alpha^2 \\ \alpha^3 & 0 \\ 0 & I_4 \end{array} \right), N_8=\left( \begin{array}{cc} \alpha & 0 \\ 0 & I_4 \\ 0 & \alpha \\ \alpha^3 & 0 \end{array} \right), N_9=\left( \begin{array}{cc} \alpha & 0 \\ 0 & \alpha^3 \\ 0 & \alpha^2 \\ I_4 & 0 \end{array} \right). \end{aligned}\]
Then clearly each column sum of \(N\) is two and \(NN^T=9\left(I_4\mathrm{\otimes }I_4\right)+\left(J_4\mathrm{-}I_4\right)\mathrm{\otimes }\left(J_4\mathrm{-}I_4\right).\) Hence \(N\) represents the incidence matrix of LS4. Further since each row sum of block matrices \(N_i(1\le i\le 9)\) is one, the design is resolvable.
The author is obliged to anonymous reviewers and Dr. M. K. Singh for their valuable suggestions in improving the content and presentation of the paper.