Semi Williamson type matrices and the W(2n, n) conjecture

Authors

Jennifer Seberry, University of WollongongFollow
Xian-Mo Zhang, University of WollongongFollow

Publication Details

Jennifer Seberry and Xian-Mo Zhang, Semi Williamson type matrices and the W(2n, n) conjecture, Journal of Combinatorial Mathematics and Combinatorial Computing, 11, (1992), 65-71.

Abstract

Four (1, -1, 0)-matrices of order m, X = (Xij), Y = (Yij), Z = (Zij), U = (Uij) satisfying

(i) XX^T + yyT + ZZ^T + UU^T = 2mI_m ,

(ii) x²_ij + y²_ij + z²_ij + U²_ij = 2, i, j = 1, ... ,m,

(iii) X, Y, Z, U mutually amicable,

will be called semi Williamson type matrices of order m. In this paper we prove that if there exist Williamson type matrices of order n₁,...n_k. then there exist semi Williamson type matrices of order N = II^k_j=1 n^r _j ^j, where r_j are non-negative integers. As an application, we obtain a W(4N,2N).

Although the paper presents no new W(4n,2n) for n, odd, n < 3000, it is a step towards proving the conjecture that there exists a W(4n, 2n) for any positive integer n. This conjecture is a sub-conjecture of the Seberry conjecture [4, page 92] that W(4n, k) exist for all k = 0,1, ... , 4n. In addition we find infinitely many new W(2n, n), n odd and the sum of two squares.