Physical Sciences and Mathematics Commons^™

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 …