Tactical decomposable regular group divisible designs and their threshold schemes

1. Introduction

A balanced incomplete block (BIB) design or a 2 − (v, k, λ) design is an arrangement of v elements into b = λ(v² − v)/(k² − k) blocks, each of size k (k < v) such that every element occurs in exactly r blocks and any two distinct elements occur together in λ blocks. Further, a BIB design is symmetric if v = b and is self-complementary if v = 2k.

1.1. Tactical decomposable design

Let a (0, 1) – matrix N have a decomposition N = [N_ij]_{i=1,…,s; j=1,…,t} 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(v, b, r, k), D is called row (column) tactical decomposable. D is called uniform row (column) tactical decomposable if r_ij = α (k_ij = β) ∀i, j. If each N_ij is an m × m matrix, D is called square tactical decomposable design, STD(m).

Several methods of constructions of tactical decomposable rectangular, group divisible and L₂–type designs may be found in Bekar et al. [2], Singh and Saurabh [19], Saurabh and Sinha [16], [17] and Saurabh [14], among others.

1.2. Group divisible design

Let v = mn elements be arranged in an m × n array. A regular group divisible (RGD) 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 × n array, occur together in λ₁ blocks whereas remaining pairs of elements occur together in λ₂ blocks; and
4. r − λ₁ > 0, rk − vλ₂ > 0.

Further let N be a v × b incidence matrix of a block design such that J_vN = kJ_v×b and satisfies the following conditions (1) or (2).

(i)\[\begin{equation} \label{GrindEQ__1_} NN’=\left(r-{\lambda }_1\right)\left(I_m\mathrm{\otimes }I_n\right)+\left({\lambda }_1-{\lambda }_2\right)\left(I_m\mathrm{\otimes }J_n\right)+{\lambda }_2\left(J_m\mathrm{\otimes }J_n\right). \end{equation}\tag{1} \]

Let \(R_i\) and \(R_j\) be any two rows of blocks of \(N\). Then from (1), their inner product is

\[\begin{align*}R_i\bullet R_j&= \begin{cases} rI_n+{\lambda }_1{\left(J-I\right)}_n,& i=j \\ {\lambda }_2J_n,& i\neq j \end{cases}\\ &=\begin{cases} (r-{\lambda }_1)I_n+{\lambda }_1J_n,&i=j ,\\ {\lambda }_2J_n,& i\neq j. \end{cases} \end{align*}\] (ii)\[\begin{equation} \label{GrindEQ__2_} NN’=\left(r-{\lambda }_2\right)\left(I_n\mathrm{\otimes }I_m\right)+{\lambda }_2\left(J_n\mathrm{\otimes }J_m\right)+\left({\lambda }_1-{\lambda }_2\right)\left\{\left(J_n\mathrm{-}I_n\right)\mathrm{\otimes }I_m\right\}. \end{equation}\tag{2}\]

Then (2) \(\Rightarrow\)

\[\begin{align*} R_i\bullet R_j&=\begin{cases} rI_m+{\lambda }_2{\left(J-I\right)}_m,& i=j, \\ {\lambda }_1I_m+{\lambda }_2{\left(J-I\right)}_m,& i\neq j, \end{cases}\\ &=\begin{cases} (r-{\lambda }_2)I_m+{\lambda }_2J_m,& i=j, \\ {(\lambda }_1-{\lambda }_2)I_m+{\lambda }_2J_m,& i\neq j. \end{cases} \end{align*}\]

Then N represents a GD design with parameters: v = mn, r, k, b, λ₁, λ₂, m, n. For GD schemes, we refer to Saurabh [14]. A GD design will be called STD(n) or STD(m) with orthogonal rows if its incidence matrix satisfies the conditions (1) or (2) respectively.

1.3. (μ₁, μ₂, …, μ_t)–resolvable design

Let the incidence matrix N of a block design D(v, r, k, b) may be decomposed into submatrices as N = (N₁ | N₂ | … | N_t) such that each row sum of N_i (1 ≤ i ≤ t) is μ_i. Then the design is (μ₁, μ₂, …, μ_t)–resolvable [see Kageyama [8], Saurabh [13]]. If μ₁ = μ₂ = … = μ_t = μ then the design is μ–resolvable. Such designs are also denoted as A–resolvable designs in combinatorial design theory [see Ge and Miao [7]]. A practical application of (μ₁, μ₂, …, μ_t)–resolvable designs may be found in Kageyama [8]. These designs may also have potential applications in coding theory and cryptography.

1.4. Cyclic and partial cyclic designs

A block design D(v, b, r, k) is cyclic if its solution may be obtained by adding the elements of a cyclic group Z_v = {0, 1, 2, …, v} mod v to the initial blocks of the design, whereas a design is partial cyclic if its solution may be obtained by developing the initial blocks under a partial cycle: 1 ↔ q, q + 1 ↔ 2q, …, [q(p − 1) + 1] ↔ v = pq of length q where (1 ↔ q) ⇐⇒ 1 → 2, 2 → 3, …, (q − 1) → q, q → 1 [see Saurabh [12]].

Some constructions of partial cyclic GD designs can be found in Dey and Nigam [6], Mukerjee et al. [10], Dey and Balasubramanian [5], Midha and Dey [9], among others.

Example 1.1. A partial cyclic solution to the GD design R80: v = 14, r = 9, k = 3, b = 42, λ₁ = 6, λ₂ = 1, m = 7, n = 2 may be obtained by developing the initial blocks: (1, 2, 8); (1, 8, 9); (1, 3, 8); (1, 8, 10); (1, 4, 8); (1, 8, 11) under a partial cycle 1 ↔ 7, 8 ↔ 14 of length 7 [see Clatworthy [4]].

Notation 1.2. I_n is the identity matrix of order n, J_v×b is the v × b matrix all of whose entries are 1 and J_v×v = J_v, A′ is the transpose of matrix A, A ⊗ B is the Kronecker product of two matrices A and B, 0_n is a zero matrix of order n × n and a (0,1)–matrix: α = circ(0 1 0 … 0) is a permutation circulant matrix of order m such that α^m = I_m. RX numbers are from Clatworthy [4].

2. Tactical decomposable RGD designs

2.1. From partial cyclic RGD designs

Proof. Let N be the incidence matrix of a RGD design D having partial cyclic solution of length m. Then the number of initial blocks is t = b/m. Our aim is to decompose N as N = [N_ij]_{i=1,…,n; j=1,…,t} where each N_ij is a square matrix of order m. Then corresponding to each initial block B_i (1 ≤ i ≤ t) of D, we obtain the i^th–column of blocks of N as follows:

Step I: Break the interval [1, mn] into n subintervals as: [1, m], [m + 1, 2m], …, [m(n − 1) + 1, mn] such that each subinterval contains m elements which is the length of partial cycle.

Step II: Let α = circ(0 1 0 … 0) be a permutation circulant matrix of order m. Then corresponding to initial block whose elements belong to above subintervals, we obtain the following m × m block matrices:

\[ \begin{align*} N_{1i}&=I_m+\alpha +{\alpha }^2+\dots +{\alpha }^{m-1}, \\ N_{2i}&=I_m+{\alpha }^{m+1}+{\alpha }^{m+2}+\dots +{\alpha }^{2m-1}\left(modm\right)=I_m+\alpha +{\alpha }^2+\dots +{\alpha }^{m-1}, \\ &\vdots \\ N_{ni}&=I_m+{\alpha }^{m(n-1)+1}+{\alpha }^{m(n-1)+2}+\dots +{\alpha }^{mn-1}\left(modm\right)=I_m+\alpha +{\alpha }^2+\dots +{\alpha }^{m-1}.\end{align*}\]

Hence we obtain a STD(m) RGD design corresponding to its partial cyclic solution. ∎

2.2. From self–complementary BIB design

Theorem 2.2. The existence of a self–complementary BIB design with parameters: \( v = 2k, r, k, b = 2r, \lambda \) implies the existence of a 3–resolvable STD (2) RGD design with parameters: \( v^* = 4k, r^* = 3r, k^* = 3k, b^* = 4r, \lambda_1 = \lambda + 2r, \lambda_2 = 2r, m = 2, n = 2k \).

Proof. Let \( M \) be the incidence matrix of a self–complementary BIB design. Then replacing \( 0 \rightarrow I_2 \) and \( 1 \rightarrow J_2 \) in \( M \), we obtain a \((0, 1)\)–matrix \( N \) such that: \( NN’ = r \left( I_{2k} \otimes I_2 \right) + 2r \left( J_{2k} \otimes J_2 \right) + \lambda \left\{ (J – I)_{2k} \otimes I_2 \right\}. \) Also, each column sum of \( N \) is \( 3k \). Hence \( N \) represents the incidence matrix of a STD (2) RGD design with above mentioned parameters.

Since the BIB design is self–complementary, we obtain \( r’ \) pairs of columns \( C_i \) and \( C_j \) such that \( C_i + C_j = J_{2k \times 1} \). Such columns will be called a pair of self–complementary columns. Further replacement of \( 0 \rightarrow I_2 \) and \( 1 \rightarrow J_2 \) in each pair of self–complementary columns yields \( r’ \) resolution classes such that each element occurs exactly three times in every class. Hence the design is 3–resolvable.

Example 2.3. Consider a selfcomplementary BIB design with parameters: v = 4, r = 3, k = 2, b = 6, λ = 1 whose incidence matrix is:

$$M=\left[ \begin{array}{cc} 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{array} \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{array} \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ 1 & 0 \\ 0 & 1 \end{array} }\right.} \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{array} \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{array} \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ 1 & 0 \\ 0 & 1 \end{array} }\right.} \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ 1 & 0 \\ 0 & 1 \end{array} \right].$$

Then replacing in \(M\), we obtain a STD (2) RGD design R164: \(v=8,r=9,k=6,b=12,{\lambda }_1=7,{\lambda }_2=6,m=2,n=4\) with incidence matrix \(N\) as given below: \[N=\left[N_1\mathrel{\left|\vphantom{N_1 N_2 N_3}\right.}N_2\mathrel{\left|\vphantom{N_1 N_2 N_3}\right.}N_3\right]=\left[ \begin{array}{cc} J_2 & I_2 \\ J_2 & I_2 \\ I_2 & J_2 \\ I_2 & J_2 \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} J_2 & I_2 \\ J_2 & I_2 \\ I_2 & J_2 \\ I_2 & J_2 \end{array} \begin{array}{cc} J_2 & I_2 \\ I_2 & J_2 \\ J_2 & I_2 \\ I_2 & J_2 \end{array} \begin{array}{cc} I_2 & J_2 \\ J_2 & I_2 \\ J_2 & I_2 \\ I_2 & J_2 \end{array} }\right.} \begin{array}{cc} J_2 & I_2 \\ I_2 & J_2 \\ J_2 & I_2 \\ I_2 & J_2 \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} J_2 & I_2 \\ J_2 & I_2 \\ I_2 & J_2 \\ I_2 & J_2 \end{array} \begin{array}{cc} J_2 & I_2 \\ I_2 & J_2 \\ J_2 & I_2 \\ I_2 & J_2 \end{array} \begin{array}{cc} I_2 & J_2 \\ J_2 & I_2 \\ J_2 & I_2 \\ I_2 & J_2 \end{array} }\right.} \begin{array}{cc} I_2 & J_2 \\ J_2 & I_2 \\ J_2 & I_2 \\ I_2 & J_2 \end{array} \right].\]

Further since each row sum of \(N_1,N_2\) and \(N_3\ \)is 3, the design is \(3-\)resolvable.

3. Illustrative examples

Here, some RGD designs of Clatworthy [4] are identified as STD \((m)\) using Theorem 2.1 where \(\alpha =circ\ (0\ 1\ 0\dots 0)\) is a permutation circulant matrix of order \(m.\) The arrangement of \(v=mn\) elements into \(m\times n\) array for following RGD designs is as follows: \[\begin{array}{ccccc} 1 & m+1 & 2m+1 & \cdots & \left(n-1\right)m+1 \\ 2 & m+2 & 2m+2 & \cdots & \left(n-1\right)m+2 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ m & 2m & 3m & \cdots & mn \end{array}\]

1) \(R80:v=14,r=9,k=3,b=42,{\lambda }_1=6,{\lambda }_2=1,m=7,n=2.\)

Initial blocks are [(1, 2, 8), (1, 8, 9), (1, 3, 8), (1, 8, 10), (1, 4, 8), (1, 8, 11)] under the partial cycles \(1\to 7,\ 8\to 14\) of length 7. Using Theorem 2.1, we have \[N=\left[N_1\mathrel{\left|\vphantom{N_1 N_2 N_3}\right.}N_2\mathrel{\left|\vphantom{N_1 N_2 N_3}\right.}N_3\right]=\left[ \begin{array}{cc} \alpha & \alpha +{\alpha }^2 \\ \alpha +{\alpha }^2 & \alpha \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} \alpha & \alpha +{\alpha }^2 \\ \alpha +{\alpha }^2 & \alpha \end{array} \begin{array}{cc} \alpha & \alpha +{\alpha }^3 \\ \alpha +{\alpha }^3 & \alpha \end{array} \begin{array}{cc} \alpha & \alpha +{\alpha }^4 \\ \alpha +{\alpha }^4 & \alpha \end{array} }\right.} \begin{array}{cc} \alpha & \alpha +{\alpha }^3 \\ \alpha +{\alpha }^3 & \alpha \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} \alpha & \alpha +{\alpha }^2 \\ \alpha +{\alpha }^2 & \alpha \end{array} \begin{array}{cc} \alpha & \alpha +{\alpha }^3 \\ \alpha +{\alpha }^3 & \alpha \end{array} \begin{array}{cc} \alpha & \alpha +{\alpha }^4 \\ \alpha +{\alpha }^4 & \alpha \end{array} }\right.} \begin{array}{cc} \alpha & \alpha +{\alpha }^4 \\ \alpha +{\alpha }^4 & \alpha \end{array} \right].\]

Then \(NN'=8\left(I_2\mathrm{\otimes }I_7\right)+\left(J_2\mathrm{\otimes }J_7\right)+5\left\{\left(J_2\mathrm{-}I_2\right)\mathrm{\otimes }I_7\right\}.\) Hence \(N\) represents the incidence matrix of \(R80\). Further since each row sum of \(N_1,N_2\) and \(N_3\ \)is 3, the design is \(3-\)resolvable.

The initial blocks of the remaining RGD designs listed below may be found in Clatworthy .

2) \(R89:v=18,r=9,k=3,b=54,{\lambda }_1=2,{\lambda }_2=1,m=9,n=2.\)

\[N=\left[N_1\mathrel{\left|\vphantom{N_1 N_2}\right.}N_2\right]=\left[ \begin{array}{cc} \alpha & \alpha +{\alpha }^3 \\ {\alpha }^7+{\alpha }^8 & {\alpha }^3 \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} \alpha & \alpha +{\alpha }^3 \\ {\alpha }^7+{\alpha }^8 & {\alpha }^3 \end{array} \begin{array}{cccc} \alpha & \alpha & \alpha & \alpha +{\alpha }^2+{\alpha }^5 \\ {\alpha }^2+{\alpha }^4 & \alpha +{\alpha }^5 & I_7+{\alpha }^6 & 0_7 \end{array} }\right.} \begin{array}{cccc} \alpha & \alpha & \alpha & \alpha +{\alpha }^2+{\alpha }^5 \\ {\alpha }^2+{\alpha }^4 & \alpha +{\alpha }^5 & I_7+{\alpha }^6 & 0_7 \end{array} \right].\]

Since each row sum of \(N_1\) and \(N_2\ \)is 3 and 6 respectively, the design is \(\left(3,\ 6\right)-\)resolvable.

3) \(R115:v=15,r=8,k=4,b=30,{\lambda }_1=6,{\lambda }_2=1,m=5,n=3.\)

\[N=\left[N_1\mathrel{\left|\vphantom{N_1 N_2}\right.}N_2\right]=\left[ \begin{array}{ccc} \alpha +{\alpha }^2 & \alpha & \alpha \\ \alpha & \alpha +{\alpha }^2 & \alpha \\ \alpha & \alpha & \alpha +{\alpha }^2 \end{array} \mathrel{\left|\vphantom{ \begin{array}{ccc} \alpha +{\alpha }^2 & \alpha & \alpha \\ \alpha & \alpha +{\alpha }^2 & \alpha \\ \alpha & \alpha & \alpha +{\alpha }^2 \end{array} \begin{array}{ccc} \alpha +{\alpha }^3 & \alpha & \alpha \\ \alpha & \alpha +{\alpha }^3 & \alpha \\ \alpha & \alpha & \alpha +{\alpha }^3 \end{array} }\right.} \begin{array}{ccc} \alpha +{\alpha }^3 & \alpha & \alpha \\ \alpha & \alpha +{\alpha }^3 & \alpha \\ \alpha & \alpha & \alpha +{\alpha }^3 \end{array} \right].\]

Since each row sum of \(N_1\ \)and \(N_2\ \)is 4, the design is \(4-\)resolvable.

4) \(R128:v=26,r=8,k=4,b=52,{\lambda }_1=0,{\lambda }_2=1,m=13,n=2.\)

\[N=\left[N_1\mathrel{\left|\vphantom{N_1 N_2}\right.}N_2\right]=\left[ \begin{array}{cc} {\alpha }^2+{\alpha }^4+{\alpha }^{10} & \alpha \\ \alpha & {\alpha }^2+{\alpha }^4+{\alpha }^{10} \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} {\alpha }^2+{\alpha }^4+{\alpha }^{10} & \alpha \\ \alpha & {\alpha }^2+{\alpha }^4+{\alpha }^{10} \end{array} \begin{array}{cc} {\alpha }^3+{\alpha }^6+{\alpha }^7 & \alpha \\ \alpha & {\alpha }^3+{\alpha }^6+{\alpha }^7 \end{array} }\right.} \begin{array}{cc} {\alpha }^3+{\alpha }^6+{\alpha }^7 & \alpha \\ \alpha & {\alpha }^3+{\alpha }^6+{\alpha }^7 \end{array} \right].\]

Since each row sum of \(N_1\ \)and \(N_2\ \)is 4, the design is \(4-\)resolvable.

5) \(R132:v=30,r=10,k=4,b=75,{\lambda }_1=2,{\lambda }_2=1,m=15,n=2.\)

\[N=\left[N_1\mathrel{\left|\vphantom{N_1 N_2 N_3}\right.}N_2\mathrel{\left|\vphantom{N_1 N_2 N_3}\right.}N_3\right]=\left[ \begin{array}{cc} \alpha +{\alpha }^3+I_{15} & {\alpha }^5 \\ {\alpha }^5 & \alpha +{\alpha }^3+I_{15} \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} \alpha +{\alpha }^3+I_{15} & {\alpha }^5 \\ {\alpha }^5 & \alpha +{\alpha }^3+I_{15} \end{array} \begin{array}{cc} \alpha +{\alpha }^5+{\alpha }^{11} & {\alpha }^2 \\ {\alpha }^2 & \alpha +{\alpha }^5+{\alpha }^{11} \end{array} \begin{array}{c} \alpha +{\alpha }^9 \\ \alpha +{\alpha }^9 \end{array} }\right.} \begin{array}{cc} \alpha +{\alpha }^5+{\alpha }^{11} & {\alpha }^2 \\ {\alpha }^2 & \alpha +{\alpha }^5+{\alpha }^{11} \end{array} \mathrel{\left|\vphantom{ \begin{array}{cc} \alpha +{\alpha }^3+I_{15} & {\alpha }^5 \\ {\alpha }^5 & \alpha +{\alpha }^3+I_{15} \end{array} \begin{array}{cc} \alpha +{\alpha }^5+{\alpha }^{11} & {\alpha }^2 \\ {\alpha }^2 & \alpha +{\alpha }^5+{\alpha }^{11} \end{array} \begin{array}{c} \alpha +{\alpha }^9 \\ \alpha +{\alpha }^9 \end{array} }\right.} \begin{array}{c} \alpha +{\alpha }^9 \\ \alpha +{\alpha }^9 \end{array} \right].\]

Since each row sum of \(N_1,N_2\) and \(N_3\ \)is 4, 4 and 2 respectively, the design is \(\left(2,\ 4,\ 4\right)-\)resolvable.

6) \(R152:v=20,r=10,k=5,b=40,{\lambda }_1=8,{\lambda }_2=1,m=5,n=4.\)

\[N=\left[N_1\mathrel{\left|\vphantom{N_1 N_2}\right.}N_2\right]=\left[ \begin{array}{cccc} \alpha +{\alpha }^2 & \alpha & \alpha & \alpha \\ \alpha & \alpha +{\alpha }^2 & \alpha & \alpha \\ \alpha & \alpha & \alpha +{\alpha }^2 & \alpha \\ \alpha & \alpha & \alpha & \alpha +{\alpha }^2 \end{array} \mathrel{\left|\vphantom{ \begin{array}{cccc} \alpha +{\alpha }^2 & \alpha & \alpha & \alpha \\ \alpha & \alpha +{\alpha }^2 & \alpha & \alpha \\ \alpha & \alpha & \alpha +{\alpha }^2 & \alpha \\ \alpha & \alpha & \alpha & \alpha +{\alpha }^2 \end{array} \begin{array}{cccc} \alpha +{\alpha }^3 & \alpha & \alpha & \alpha \\ \alpha & \alpha +{\alpha }^3 & \alpha & \alpha \\ \alpha & \alpha & \alpha +{\alpha }^3 & \alpha \\ \alpha & \alpha & \alpha & \alpha +{\alpha }^3 \end{array} }\right.} \begin{array}{cccc} \alpha +{\alpha }^3 & \alpha & \alpha & \alpha \\ \alpha & \alpha +{\alpha }^3 & \alpha & \alpha \\ \alpha & \alpha & \alpha +{\alpha }^3 & \alpha \\ \alpha & \alpha & \alpha & \alpha +{\alpha }^3 \end{array} \right].\]

Since each row sum of \(N_1\ \)and \(N_2\ \)is 5, the design is \(5-\)resolvable.

7) \(R159:v=35,r=10,k=5,b=70,{\lambda }_1=2,{\lambda }_2=1,m=5,n=7.\)

\[N=\left[N_1\mathrel{\left|\vphantom{N_1 N_2}\right.}N_2\right]=\left[circ\ \left(0_5,0_5,0_5,\ \alpha ,\alpha ,\alpha ,{\alpha }^2+I_5\mathrel{\left|\vphantom{0_5,0_5,0_5,\ \alpha ,\alpha ,\alpha ,{\alpha }^2+I_5 circ\ (0_5,0_5,0_5,\ \alpha ,\alpha ,\alpha ,{\alpha }^3+{\alpha }^4}\right.}circ\ (0_5,0_5,0_5,\ \alpha ,\alpha ,\alpha ,{\alpha }^3+{\alpha }^4\right)\right].\]

Since each row sum of \(N_1\ \)and \(N_2\ \)is 5, the design is \(5-\)resolvable.

8) \(R160:v=39,r=10,k=5,b=78,{\lambda }_1=2,{\lambda }_2=1,m=13,n=3.\) \[\begin{aligned} N&=\left[N_1\mathrel{\left|\vphantom{N_1 N_2}\right.}N_2\right]\\ &=\left[ \begin{array}{ccc} {\alpha }^2+{\alpha }^4+{\alpha }^{10} & \alpha & \alpha \\ \alpha & {\alpha }^2+{\alpha }^4+{\alpha }^{10} & \alpha \\ \alpha & \alpha & {\alpha }^2+{\alpha }^4+{\alpha }^{10} \end{array} \mathrel{\left|\vphantom{ \begin{array}{ccc} {\alpha }^2+{\alpha }^4+{\alpha }^{10} & \alpha & \alpha \\ \alpha & {\alpha }^2+{\alpha }^4+{\alpha }^{10} & \alpha \\ \alpha & \alpha & {\alpha }^2+{\alpha }^4+{\alpha }^{10} \end{array} \begin{array}{ccc} {\alpha }^3+{\alpha }^6+{\alpha }^7 & \alpha & \alpha \\ \alpha & {\alpha }^3+{\alpha }^6+{\alpha }^7 & \alpha \\ \alpha & \alpha & {\alpha }^3+{\alpha }^6+{\alpha }^7 \end{array} }\right.} \begin{array}{ccc} {\alpha }^3+{\alpha }^6+{\alpha }^7 & \alpha & \alpha \\ \alpha & {\alpha }^3+{\alpha }^6+{\alpha }^7 & \alpha \\ \alpha & \alpha & {\alpha }^3+{\alpha }^6+{\alpha }^7 \end{array} \right]. \end{aligned}\]

Since each row sum of \(N_1\ \)and \(N_2\ \)is 5, the design is \(5-\)resolvable.

9) \(R189:v=b=24,r=k=8,\ {\lambda }_1=4,{\lambda }_2=2,m=4,n=6.\)

\[N=\left[I_6\otimes \left({\alpha }^2+{\alpha }^3+I_4\right)+{\left(J-I\right)}_6\otimes \alpha \right].\]

10) \(R200:v=b=28,r=k=9,\ {\lambda }_1=5,{\lambda }_2=2,m=4,n=7.\)

\[N=\left[I_7\otimes \left({\alpha }^2+{\alpha }^3+I_4\right)+{\left(J-I\right)}_7\otimes \alpha \right].\]

11) \(R208:v=b=32,r=k=10,\ {\lambda }_1=6,{\lambda }_2=2,m=4,n=8.\)

\[N=\left[I_8\otimes \left({\alpha }^2+{\alpha }^3+I_4\right)+{\left(J-I\right)}_8\otimes \alpha \right].\]

4. Application in \(\left({2}{,\ }{n}\right){-}\)threshold schemes

Let \(\mathcal{K}\) be a finite key space and \(P\) be a finite set of participants. In a secret sharing scheme, a special participant \(D\notin P\), called the dealer, secretly chooses a key \(K\in \mathcal{K}\) and distributes one share or shadow from the share set \(S\) to each participant in a secure manner, so that no participant knows the shares given to other participants. A \(\left(t,\ n\right)-\)threshold scheme is a secret sharing scheme in which if any \(t(\le n)\) or more participants pool their shares, where \(n=\left|P\right|\), then they can reconstruct the secret key \(K\in \mathcal{K}\), but any \(n-1\) or fewer participants can gain no information about it.

According to Time Magazine (May 4, 1992, p. 13), control of nuclear weapons in Russia in early 1990s depended upon “two–out–of–three” access mechanism. The three parties involved were the President, the Defense–minister and the Defense Ministry. This would correspond to a threshold scheme with \(n=3\), \(t=2\), op. cit. Stinson and Vanstone [21], Stinson [20].

Pieprzyk and Zhang [11] obtained ideal (t, w) \(-\)threshold schemes from \(b^t\times (n+1)\) orthogonal array \(OA\ (b^t,\ n+1,\ b,\ t)\) by considering \(OA\ (i,j)\) as the shares of participants \(P_j\ (1\le j\le n)\) and \(OA\ \left(i,0\right)\) as a secret key \((1\le i\le b^t)\) where \(OA\ (i,j)\) denotes the entry in the \(i^{th}\) row and \(j^{th}\) column of \(OA\ (b^t,\ n+1,\ b,\ t)\). Stinson and Vanstone [21] obtained perfect threshold schemes from Steiner system \(S(t,\ w,\ v)\). Adachi and Lu [1] constructed \(\left(3,\ 3\right)-\)threshold schemes from magic cubes by considering magic cube as a secret key and the corresponding three cubes as the shadows.

Some recent constructions of perfect secret sharing schemes from doubly resolvable GD designs and orthogonal resolutions of certain combinatorial designs can be found in Saurabh and Sinha [15,18]. A recent survey on threshold schemes from combinatorial designs may be found in Bose [3].

Present scheme: Consider a STD \((m)\) RGD design whose each submatrix is of size \(m\). Then there are \(n\) rows of blocks in its incidence matrix \(N.\) The dealer provides rows \(R_i(1\le i\le n)\) of blocks of \(N\) to \(n\) participants as their shares. Two participants can reveal the secret if their shares \(R_i\) and \(R_j\) are orthogonal rows of the STD \((m)\) RGD design, i.e., \[\begin{aligned} R_i\bullet R_j&=\begin{cases} rI_m+{\lambda }_2{\left(J-I\right)}_m,& i=j, \\ {\lambda }_1I_m+{\lambda }_2{\left(J-I\right)}_m,& i\neq j, \end{cases}\\ &=\begin{cases} (r-{\lambda }_2)I_m+{\lambda }_2J_m,& i=j, \\ {(\lambda }_1-{\lambda }_2)I_m+{\lambda }_2J_m,& i\neq j. \end{cases} \end{aligned}\]

Hence corresponding to a STD \((m)\) RGD design, we obtain a \(\left(2,\ n\right)-\)threshold scheme.

Further using Theorem 2.1 and 2.2, we obtain:

Scheme: The tactical decomposable RGD designs in Theorem 2.1 correspond to \(\left(2,\ n\right)-\)threshold schemes whereas the designs of Theorem 2.2 correspond to \(\left(2,\ 2k\right)-\) threshold schemes.

Example 4.1. The STD (4) RGD design R200 given in Section 3 can be used to obtain a \(\left(2,\ 7\right)-\) threshold scheme.