Physical Sciences and Mathematics Commons^™

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, …

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, …

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 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.

Some Results On Configurations, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

A (v, k, lambda) configuration is conjectured to exist for every v, k and lambda satisfying lambda(v-l) = k(k-l) and k - lambda is a square if v is even, x2 = (k - lambda)y2+(-1)(v-1)/2lamdaZ2 has a solution in integers x,y and z not all zero for v odd.

Integer Matrices Obeying Generalized Incidence Equations, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

We consider integer matrices obeying certain generalizations of the incidence equations for (v, k, lambda)-configurations and show that given certain other constraints, a constant multiple of the incidence matrix of a (v, k, lambda)-configuration may be identified as the solution of the equation.

(V, K, Lambda)-Configurations And Hadamard Matrices, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

(v, k, lambda) Configirations and Hadamard matrics

Hadamard Designs, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

In this paper it is shown that an Hadamard design with each letter repeated once and only once can exist for 2, 4 and 8 letters only. L.D. Baumert and Marshall Hall, Jr have found a design with four letters each repeated three times. Their design and the design on four letters each repeated once, found by J. Williamson, is the totality previously published.

A Note Of A Class Of Hadamard Matrices, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

An Hadamard matrix H is a matrix of order n all of whose elements are + 1 or -1 and which satisfies H ffT = nIn . H = S + In is a skew-type Hadamard matrix if ST = -So It is conjectured that an Hadamard matrix always exists for n = 4t, t any integer. Many known matrices and classes of matrices can be found in [1].

Two New Block Designs, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

In this note the matrices W, X. Y, and Z are the incidence matrices of the (u, k, ,\,) configurations (15, 7, 3), (25,9,3), (45, 12,3), and (36, 15,6), respectively. Wand X are new formulations of these configurations and Yand Z were previously not known (see [I, pp. 295, 297]).

A Class Of Hadamard Matrices, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Whenever there exists a quasi-skew Hadamard matrix of order 4m and (4n - I, k, m - n + k) and (4n - I, u, u - n) configurations with circulant incidence matrices, then there exists an Hadamard matrix of order 4m(4n - I).

Equivalence Of Hadamard Matrices, W. D. Wallis, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Suppose In is a square-free odd integer, and A and B are any two Hadamard matrices of order 4m. We will show that A and B are equivalent over the integers (that is, B can be obtained from A using elementary row and column operations which involve only integers).