Physical Sciences and Mathematics Commons^™

Construction Of Williamson Type Matrices, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Recent advances in the construction of Hadamard matrices have depended on the existence of Baumert-Hall arrays and four (1, - 1) matrices A, B, C, D of order m which are of Williamson type, that is they pair-wise satisfy

i) MN^T = NM^T, M, N E {A, B, C, D} and

ii) AA^T + BB^T + CC^T + DD^T = 4mI_m.

It is shown that Williamson type matrices exist for the orders m = s(4s - 1), m = s(4s + 3) for s E {1, 3, 5, ... ,25} and …

Orthogonal Designs, Anthony V. Geramita, Joan Murphy Geramita, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Orthogonal designs of special type have been extensively studied, and it is the existence of these special types that has motivated our study of the general problem of the existence of orthogonal designs.

This paper is organized in the following way. In the first section we give some easily obtainable necessary conditions for the existence of orthogonal designs of various order and type. In Section 2 we briefly survey the examples of such designs that we have found in the literature. In the third section we describe several methods for constructing orthogonal designs. In the fourth section we obtain some …

Construction Of Amicable Orthogonal Designs, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Infinite families of amicable orthogonal designs are constructed with

(i) both of type (1, q) in order q + 1 when q = 3, (mod 4 ) is a prime power,

(ii) both of type (1, q) in order 2(q+1) where q = 1 (mod 4) is a prime power or q + 1 is the order of a conference matrix,

(iii) both of type (2, 2q) in order 2(q+l) when q = 1 (mod 4) is a prime power or q + 1 is the order of a conference matrix.

On The Matrices Used To Construct Baumert-Hall Arrays, Richard B. Lakein, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Four circulant (or type 1) (0,1,-1) matrices X₁, X₂, X₃, X₄ of order t with the property that each of the t² positions is non-zero in precisely one of the X_i and such that

X₁X₁^T+ X₂X₂^T + X₃X₃^T + X₄X₄^T = tI_t

will be called T-matrices.

This paper studies the construction, use and properties of T-matrices giving a new construction for Hadamard matrices and some new equivalence results for Hadamard matrices and Baumert-Hall …

An Algorithm For Orthogonal Designs, Peter Eades, Peter J. Robinson, Jennifer Seberry, Ian S. Williams

Faculty of Informatics - Papers (Archive)

Let A =(s_i) be an n-tuple of positive integers such that Es_i = 2^k. We give an algorithm which shows that there exists a p = (R_A(n, k) - (k+1)) such that there is an orthogonal design of type (2^ps₁, 2^ps₂,..., 2^ps_n) in order 2^k+p. We evaluate the maximum of p over n-tuples A which add to 2^k. Hence we deduce that for any n and k there is an integer q = max R_A(n, k) - …

Families Of Weighing Matrices, Anthony V. Geramita, Norman J. Pullman, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

A weighing matrix is an n x n matrix W = W(n, k) with entries from {0, 1, -l}, satisfying WW^t = kI_n. We shall call k the degree of W. It has been conjectured that if n = 0 (mod 4) then there exist n x n weighing matrices of every degree k < n.

We prove the conjecture when n is a power of 2. If n is not a power of two we find an integer t < n for which there are weighing matrices of every degree

A Note On Supplementary Difference Sets, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Let S₁, S₂,···, S_n be subsets of G, a finite abelian group of order v, containing k₁, k₂,...,kn elements respectively. Write T_i for the totality of all differences between elements of S_i (with repetitions), and T for the totality of elements of all the T_i. We will denote this by T= T₁ & T₂ & ... & T_n. If T contains each non-zero element of G a fixed number of times, lambda say, then the sets S₁, S₂, ..., …

Orthogonal Designs Iii: Weighing Matrices, Anthony V. Geramita, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

A weighing matrix W = W(n,k) of order n and weight k is a square (0,l,-l)-matrix satisfying

WW^t -kI_n

An orthogonal design of order n on a single variable is a weighing matrix and consequently the study of orthogonal designs is intimately connected with the study of weighing matrices.

This paper reviews and updates the current status of the conjectures:

I. Let n = 2 (mod 4). Then there exists a W(n,k) if and only if k < n - 1 is the sum of two integer squares;

II. Let n = 0 (mod 4). Then there exists a W(n,k) for each k < n. This conjecture has been verified for n = 28, 2^t+l, 2^{t +l}·3 …

Williamson Matrices Of Even Order, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Recent advances in the construction of Hadamard matrices have depended on the existence of Baumert-Hall arrays and Williamson-type matrices. These latter are four (1,-1) matrices A,B,C,D, of order m, which pairwise satisfy

(i) MN^T = NM^T, M,N E (A,B,C,D),

and (ii) AA^T +BB^T +CC^T +DD^T = 4mI_m, where I is the identity matrix.

Currently Williamson matrices are known to exist for all orders less than 100 except: 35,39,47,53,59,65,67,70,71,73,76,77,83,89,94.

This paper gives two constructions for Williamson matrices of even order, 2n. This is most significant when no Williamson matrices of order n …

A Survey Of Orthogonal Designs, Anthony V. Geramita, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

This paper surveys orthogonal designs which are an overview of Baumert-Hall arrays, Hadamard matrices and weighing matrices.

The known results are given and unsolved problems indicated.

Kronecker Products And Bibds, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Recursive constructions are given which permit, under conditions described in the paper, a (v, b, r, k, lambda)-configuration to be used to obtain a (v', b', r', k, lambda)-configuration.

Although there are many equivalent definitions we will mean by a (v, b, r, k, lambda)-configuration or BIBD that (0, 1)-matrix A of size v x b with row sum r and column sum k satisfying

AA^T = (r - lambda)I + lambdaJ

where, as throughout the remainder of this paper, I is the identity matrix and J the matrix with every element +1 whose sizes should be determined from …

A Note On Amicable Hadamard Matrices, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

The existence of Szekeres difference sets, X and Y, of size 2f with y E Y = -y E Y, where q = 4f + 1 is a prime power, q = 5 (mod 8) and q = p2 + 4, is demonstrated. This gives amicable Hadamard matrices of order 2(q + 1), and if 2q is also the order of a symmetric conference matrix, a regular symmetric Hadamard matrix of order 4q² with constant diagonal.

Hadamard Matrices Of Order 28m, 36m, And 44m, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

We show that if four suitable matrices of order m exist then there are Hadamard matrices of order 28 m, 36 m, and 44 m. In particular we show that Hadamard matrices of orders 14(q + 1), 18(q + 1), and 22(q + 1) exist when q is a prime power and q = l(mod 4).

Also we show that if n is the order of a conference matrix there is an Hadamard matrix of order 4mn. As a consequence there are Hadamard matrices of the following orders less than 4000:

476, 532, 836, 1036, 1012, 1100, 1148, 1276, 1364, …

Recent Advances In The Construction Of Hadamard Matrices, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

In the past few years exciting new discoveries have been made in constructing Hadamard matrices. These discoveries have been centred in two ideas:

(i) the construction of Baumert-Hall arrays by utilizing a construction of L. R. Welch, and

(ii) finding suitable matrices to put into these arrays.

We discuss these results, many of which are due to Richard J. Turyn or the author.

A List Of Balanced Incomplete Block Designs For R < 30, Jane W. Di Paola, Jennifer Seberry, W D. Wallis

Faculty of Informatics - Papers (Archive)

A balanced incomplete block design consists of a set of v elements arranged into b k-element subsets called blocks such that each element occurs r times and each pair of elements appears in lambda distinct blocks. The numbers v,b,r,k,lambda are called the parameters of the design. A necessary condition that a design exist is that the parameters be integers satisfying:

(1) vr = bk

( 2) r(k-1) = lambda (v-1)

Families Of Codes From Orthogonal (0,1,-1)-Matrices, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Sloane and Seidel have constructed (n,2n,¹/₂(n-2)) and (n-1,2n,¹/₂(n-2)) codes whenever n = 1 + a² + b² = 2(mod 4), a,b integer, is the order of a conference matrix. We give constructions for (n,2n,¹/₂(n-2)) and (n-1,2n,¹/₂(n-4)) codes when n = 2(mod 4) and conference matrices cannot exist.

In particular we give results for n = 22, 34, 66, 70, 106,130,154,162,202,210, ... ,"210, ... , but our codes are not as ""good" as those from Hadamard matrices or of Sloane and Seidel".

Some Matrices Of Williamson Type, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Recent advances in the construction of Hadamard matrices have depended on the existence of Baumert-Hall arrays and four (1,-1) matrices A, B, C, D of order m which are of Williamson type; that is, they pairwise satisfy

(i) MN^T = NM^T, and

(ii) AA^T + BB^T + CC^T + DD^T = 4mI_m

We show that if p = 1 (mod 4) is a prime power then such matrices exist for m = ¹/₂p(p+1). The matrices constructed are not circulant and need not be symmetric. This means there are Hadamard …

A Note On Bibds, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

A balanced incomplete block design or BlBD is defined as an arrangement of v objects in b blocks, each block containing k objects all different, so that there are r blocks containing a given object and lambda blocks containing any two given objects.

In this note we shall extend a method of Sprott [2, 3] to obtain several new families of BIBD's. The method is based on the first Module Theorem of Bose [1] for pure differences.

We shall frequently be concerned with collections in which repeated elements are counted multiply, rather than with sets. If T₁ and T …

Complex Hadamard Matrices, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

R. J. Turyn introduced complex Hadamard matrices and showed that if there is a complex Hadamard matrix of order c and a real Hadamard matrix of order h > 1, then there is a real Hadamard matrix of order hc.

Previously, complex Hadamard matrices were only known for a few small orders and the orders for which symmetric conference matrices were known. These latter are known only to exist for orders which can be written as 1 + a² + b² where a, b are integers.

We give many constructions for new infinite classes of complex Hadamard matrices and …

Some Remarks On Supplementary Difference Sets, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Let S_1,S₂ ,... ,S_n be subsets of V, a finite abelian group of order v written in additive notation, containing k₁ k₂,... ,k_n elements respectively. Write T_i for the totality of all differences between elements of S_i (with repetitions), and T for the totality of elements of all the T_i. If T contains each non-zero element of V a fixed number of times, lambda say, then the sets S₁, S₂,... ,S_n will be called n - {v; k₁, k₂, …

Some Classes Of Hadamard Matrices With Constant Diagonal, Jennifer Seberry, Albert Leon Whiteman

Faculty of Informatics - Papers (Archive)

The concepts of circulant and back circulant matrices are generalized to obtain incidence matrices of subsets of finite additive abelian groups. These results are then used to show the existence of skew-Hadamard matrices of order 8(4f+l) when f is odd and 8f + 1 is a prime power. This shows the existence of skew-Hadamard matrices of orders 296, 592, 1184, 1640, 2280, 2368 which were previously unknown.

A Construction For Hadamard Arrays, Joan Cooper, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

We give a construction for Hadamard arrays and exhibit the arrays of orders 4t , tE{l,3,5,7, ... 19} This gives seventeen new Hadamard matrices of order less than 4000.

Orthogonal (0,1,-1) Matrices, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

We study the conjecture: There exists a square (0,l,-l)-matrix W = W(w,k) of order w satisfying

WW^T= kI_w

for all k = 0, 1,..., w when w = 0 (mod 4). We prove the conjecture is true for 4, 8, 12, 16, 20, 24, 28, 32, 40 and give partial results for 36, 44, 52, 56.

On Integer Matrices Obeying Certain Matrix Equations, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

We discuss integer matrices B of odd order v which satisfy

Br = ± B, BBr = vI - J, BJ = O.

Matrices of this kind which have zero diagonal and other elements ± 1 give rise to skew-Hadamard and n-type matrices; we show that the existence of a skew-Hadamard (n-type) matrix of order h implies the existence of skew-Hadamard (n-type) matrices of orders (h - 1)⁵ + 1 and (h - 1)⁷ + 1. Finally we show that, although there are matrices B with elements other than ± 1 and 0, the equations force considerable restrictions …

Cyclotomy, Hadamard Arrays And Supplementary Difference Sets, David C. Hunt, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

A 4n x 4n Hadamard array, H, is a square matrix of order 4n with elements ± A, ± B, ± C, ± D each repeated n times in each row and column. Assuming the indeterminates A, B, C, D commute, the row vectors of H must be orthogonal. These arrays have been found for n = 1 (Williamson, 1944), n = 3 (Baumert-Hall, 1965), n = 5 (Welch, 1971), and some other odd n < 43 (Cooper, Hunt, Wallis).

The results for n = 25, 31, 37, 41 are presented here, as is a result for n = 9 not based on supplementary difference …

On Supplementary Difference Sets, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Given a finite abelian group V and subsets S₁, S₂, ... ,S_n of V, write T_i for the totality of all the possible differences between elements of S_i (with repetitions counted multiply) and T for the totality of members of all the T_i. If T contains each non-zero element of V the same number of times, then the sets S_1, S₂,...,S_n will be called supplementary difference sets.

We discuss some properties for such sets, give some existence theorems and observe their use in the construction of Hadamard …

A Skew-Hadamard Matrix Of Order 92, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Previously the smallest order for which a skew-Hadamard matrix was not known was 92. We construct such a matrix below.

Some (1, -1) Matrices, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

We define an n-type (1, -1) matrix N = 1 + R of order n ~ 2 (mod 4) to have R symmetric and R2 = (n - 1)/n. These matrices are analogous to skewtype matrices M = 1 + W which have W skew-symmetric.

Amicable Hadamard Matrices, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

If X is a symmetric Hadamard matrix, Y is a skew-Hadamard matrix, and XYT is symmetric, then X and Y are said to be amicable Hadamard matrices. A construction for amicable Hadamard matrices is given, and then amicable Hadamard matrices are used to generalize a construction for skew-Hadamard matrices.

Combinatorial Matrices, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

We investigate the existence of integer matrices B satisfying the equation BBT = rI + sJ where T denotes transpose, r and s are integers, I is the identity matrix and J is the matrix with every element +1.