Block Structured Hadamard matrices from certain arrays

: We have constructed Block structured Hadamard matrices in which odd number of blocks are used in a row (column). These matrices are di ff erent than those introduced by Agaian. Generalised forms of arrays developed by Goethals-Seidel, Wallis-Whiteman and Seberry-Balonin heve been employed. Such types of matrices are applicable in the constructions of nested group divisible designs.


Introduction
Hadamard matrices discovered by Sylvester in 1867 [1] have profound structural properties.These kind of matrices have Hadamard matrices as their sub blocks.In his matrices even number of Hadamard blocks were arranged in a row (or column).However in 1985 Agaian [2] introduced Hadamard matrices with odd number of Hadamard blocks in a row.These matrices are named Block Structured Hadamard matrices.He has demonstrated their application in signal processing as well [3].The matrices constructed by him are in correspondence to the Williamson matrices arranged in his array.
Williamson's array has been further generalized by Goethals and Seidel [4].Matrices suitable for Goethals-Seidel arrays have been constructed by several authors [5][6][7].In 2015, Seberry and Balonin [8] introduced what is known as the "propus array" and its generalizations.This was accomplished by imposing certain restrictions on Williamson's array and applying the arrangement principles from the Goethals-Seidel array.
The result presented in this paper involves the construction of block-structured Hadamard matrices using the generalizations of Goethals-Seidel array and Seberry-Balonin array.Similar to that of Agaian, these matrices also have odd number of blocks arranged in a row.However these are different from Agaian's matrices, as these do not correspond to Williamson matrices.Furthermore, their applications in nested group divisible design are also discussed.
The paper is organized as follows: Section 2 contains all the relevant definitions and results.Section 3 comprises of the main result of this paper.Results are presented in three main theorems and two corollaries.Examples of each type are also included here.Section 4 is conclusion in which we have discussed some properties and an application of block structured Hadamard matrices in the construction of nested group divisible designs.

Preliminaries
We recall some basic definitions here [4,[9][10][11][12].An Hadamard matrix is a square matrix H of order n with entries ±1 such that HH ⊤ = nI n .Four matrices A, B, C, D of order n with entries ±1 are called Williamson matrices if (i) A, B, C, D are circulant, symmetric and commuting (ii)A 2 + B 2 + C 2 + D 2 = 4nI n .If A, B, C, D of same order are circulant ±1 matrices satisfying AA ⊤ + BB ⊤ + CC ⊤ + DD ⊤ = 4nI n then these are called Goethals-Seidel (GS) matrices.Dokovic et al. [6] have constructed GS matrices of type (ssss), (sssk), (sskk), (skkk) where 's' stand for symmetric and 'k' for skew type GS matrices.In this article we shall be using (sssa) and (kkka) type GS matrices, where 'a' stands for any (symmetric, skew type or other) type of matrix.An Hadamard matrix H of order 4tn will be called Block Structured if all its 4t × 4t blocks are Hadamard matrices for a given t.If A = [a i j ] and B = [b i j ] be two matrices of order m and n respectively then the Kronecker product A ⊗ B is the matrix of order mn given by A ⊗ B = [a i j B].Two matrices A and B are said to be Amicable if AB ⊤ = BA ⊤ .Let the elements z i of an additive abelian group G be ordered in a fixed way.Let X ⊂ G. Then the matrix M = (m i j ) defined by is called type 1 incidence matrix of X in G, and the matrix N = (n i j ) defined by 0 otherwise, is called type 2 incidence matrix of X in G.A Circulant matrix M = (m i j ) defined by m i j = m 1, j−i+1 is a special case of type 1 matrix and a backcirculant matrix N = (n i j ) defined by n i j = n 1,i+ j−1 is a special case of type 2 matrix.Following proposition is useful in development of the results in this paper.
Proposition 1. [Seberry] If X and Y are type 1 matrices and Z is type Notation: Throughout this paper I n denotes the identity matrix of order n and J n denotes the all-1 matrix of order n.Wherever I and J are used without subscript, they have the same meaning and their orders can be determined from the context.

Main Result
Lemma 1.Let A i , i = 0, 1, 2, 3 be circulant matrices of order n and let R = [r i j ] be defined as r i,n−i+1 = 1, r i j = 0 otherwise.Then (1). □ Using this lemma following Theorem can be proven easily.
Then we have (1) If there exist Goethals-Seidel matrices A i , i = 0, 1, 2, 3 of order n of the type (kkka) or (sssa) then there exists a block structured Hadamard matrix of order 4nt with Hadamard blocks of order 4t.
(2) If there exist type 2 matrix A 0 and three type 1 matrices A i ; 1 ≤ i ≤ 3 defined on the same Abelian group of order n such that 3 i=0 A i A ⊤ i = 4nI n then there exists a block structured Hadamard matrix of order 4nt with Hadamard blocks of order 4t.

Proof. (1) Define a matrix H by
where A i s correspond to 's' or 'k' type for i = 1, 2, 3 and A 0 corresponds to 'a' type.Then Hence H is an Hadamard matrix.Now each 4t ⊗ 4t block (say H i j ) is a linear combination of X ′ i s and Y ′ i s of the form Case 1: A i s are of the type (kkka) Then Case 2: A i s are of the type (sssa) Then ⇒ each 4t ⊗ 4t block of H is an Hadamard matrix.
(2) Proof is similar to case (1), just define H by

Journal of Combinatorial Mathematics and Combinatorial Computing
Volume 117, 169-175 together with (sssa) or (kkka) type Goethals-Seidel matrices of order n taken as A s i give the Block Structured Hadamard matrix of order 8n with Hadamard blocks of order 8.
Theorem 2. Let there exist three (0, 1, −1)-matrices of order 4t (t Then we have (1) If there exist four Williamson matrices A i , 1 ≤ i ≤ 4 such that A 4 = A 2 of order n then there exists a block structured Hadamard matrix of order 4nt with Hadamard blocks of order 4t.
(2)If there exist three pairwise amicable ±1 matrices A i ; 1 then there exists a block structured Hadamard matrix of order 4nt with Hadamard blocks of order 4t.
Proof.Define new X i as If there exist matrices X i , i = 1, 2, 3 of order 4t, t ∈ N satisfying conditions of Theorem (2), and there exists three Amicable Hadamard matrices of order r then there exist X i matrices of order 4rt.
Proof.Let H i , i = 1, 2, 3 be Amicable Hadamard matrix of order r.Define new X i as X i ⇒ X i ⊗ H i , i = 1, 2, 3. □ Theorem 3. Let there exist (0, 1, −1)-matrices X i , Y j ; Then we have, if there exist three pairwise commutative ±1 matrices A i ; 0 ≤ i ≤ 2 of order n such that A 4nI n then there exists a block structured Hadamard matrix of order 4nt with Hadamard blocks of order 4t.

Conclusion
In this paper we have constructed block structured Hadamard matrices different from those of Sylvester and Agaian.Matrices X ′ i s and Y ′ i s of Theorem (1) can be obtained from Goethals-Seidel array and its generalization.In this method block structured Hadamard matrix of order 4nt is constructed using the blocks of a set containing at the most 16 distinct Hadamard blocks of order t.(Here two Hadamard blocks H i and H j are considered to be distinct if H i ±H j .)Matrices A ′ i s can be found in the works of several authors specially in [4,11,12].Matrices X ′ i s and Y ′ i s of Theorem (2) can be obtained from Propus array and Propus type arrays.Seberry and Balonin have constructed required A ′ i s matrices in abundance [8].Methods of construction of these matrices are not proposed here.In this construction there are precisely four distinct Hadamard blocks viz.±(X 1 + X 2 + X 3 ), ±(X 1 − X 2 + X 3 ), ±(X 1 − X 2 − X 3 ) and ±(X 1 + X 2 − X 3 ).If the matrices A ′ i s are constructed from Turyn's method then there are only three distinct Hadamard blocks viz.±(X 1 + X 2 + X 3 ), ±(X 1 − X 2 + X 3 ), ±(X 1 − X 2 − X 3 ).Theorem (3) is a product of the Theorems (1) and (2).In this construction number of distinct Hadamard blocks are 16.Efforts could be made to find a method to construct matrices X ′ i s and Y ′ i s in general, which could result in some Hadamard matrices of new orders.Block structured Hadamard matrices may be used in nested group divisible (GD) designs as follows: It is well known that replacing 1 by I 2 and −1 by (J − I) 2 in a Hadamard matrix H of order 4nt we obtain a series of resolvable semi-regular GD designs with parameters (see Saurabh and Sinha [13]): ( Further since H contains Hadamard blocks each of order 4t, removing s rows of blocks of incidence matrix of (1) we obtain a series of GD designs with parameters: v = 8t(n − s), b = 8nt, r = 4nt, k = 4t(n − s), λ 1 = 0, λ 2 = 2nt, m * = 4t(n − s), n * = 2; s < n. (2) Hence a GD design with parameters (2) is nested within a GD design with parameters (1).For details on GD designs we refer to Raghvarao and Padgett [14], Saurabh and Sinha [13] and Saurabh and Prasad [15].