Physical Sciences and Mathematics Commons^™

Some Results On Weighing Matrices, Jennifer Seberry, Albert Leon Whiteman

Faculty of Informatics - Papers (Archive)

It is shown that if q is a prime power then there exists a circulant weighing matrix of order q² + q + 1 with q² non-zero elements per row and column.

This result allows the bound N to be lowered in the theorem of Geramita and Wallis that " given a square integer k there exists an integer N dependent on k such that weighing matrices of weight k and order n and orthogonal designs (1, k) of order 2n exist for every n > N".

On Hadamard 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 pairwise satisfy

(i) MN^T = NM^T and

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

If (i) is replaced by (i')MN = NM we have Goethals-Seidel matrices. These matrices are very important to the determination of the Hadamard conjecture: that there exists an Hadamard matrix of order 4t for all natural numbers t. This paper …

Orthogonal Designs: Ii, Anthony V. Geramita, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

Orthogonal designs are a natural generalization of the Baumert-Hall arrays which have been used to construct Hadamard matrices. We continue our investigation of these designs and show that orthogonal designs of type (1, k) and order n exist for every k < n when n = 2^t+2. 3 and n = 2^t+2.5 (where t is a positive integer). We also find orthogonal designs that exist in every order 2n and others that exist in every order 4n.

Coupled with some results of earlier work, this means that the weighing matrix conjecture 'For every order n = 0 (mod4) there is, for each …

Orthogonal Designs Iv: Existence Questions, Anthony V. Geramita, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

In [5] Raghavarao showed that if n = 2 (mod 4) and A is a {O, 1, -1} matrix satisfying AA^t = (n - 1) I_n. then n - 1 = a² - b² for a, b integers. In [4] van Lint and Seidel giving a proof modeled on a proof of the Witt cancellation theorem, proved more generally that if n is as above and A is a rational matrix satisfying AA^t = kI_n then k = q_1² + q₂² (q₁, q₂ E Q, the …

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) - …