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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, …