Some matrices of Williamson type

Authors

Jennifer Seberry, University of WollongongFollow

Publication Details

Jennifer Seberry Wallis, Some matrices of Williamson type, Utilitas Mathematica, 4, (1973), 147-154.

Abstract

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 matrices of order 2p(P+1)t and 10p(p+1)t for t E {1,3,5,...,59} u {1 + 2^a 10^b26^c ,a,b,c non-negative integers} , which is a new infinite family.