Abstract

Suppose A1,..., As are (1,  1) matrices of order m satisfying {Mathematical expression} {Mathematical expression} {Mathematical expression} {Mathematical expression} Call A1,..., As, a regular s set of matrices of order m if Eq. 13 are satisfied and a regular sset of regular matrices if Eq. 4 is also satisfied, these matrices were first discovered by J. Seberry and A.L. Whiteman in "New Hadamard matrices and conference matrices obtained via Mathon's construction", Graphs and Combinatorics, 4(1988), 355377. In this paper, we prove that (i) if there exist a regular sset of order m and a regular tset of order n there exists a regular sset of order mn when t =sm (ii) if there exist a regular sset of order m and a regular tset of order n there exists a regular sset of order mn when 2t = sm (m is odd) (iii) if there exist a regular sset of order m and a regular tset of order n there exists a regular 2sset of order mn when t = 2sm As applications, we prove that if there exist a regular sset of order m there exists (iv) an Hadamard matrices of order 4hm whenever there exists an Hadamard matrix of order 4h and s =2h (v) Williamson type matrices of order nm whenever there exists Williamson type matrices of order n and s = 2n (vi) an OD(4mp;ms1,...,msu whenever an OD (4p;s1,...,su)exists and s = 2p (vii) a complex Hadamard matrix of order 2cm whenever there exists a complex Hadamard matrix of order 2c and s = 2c This paper extends and improves results of Seberry and Whiteman giving new classes of Hadamard matrices, Williamson type matrices, orthogonal designs and complex Hadamard matrices. © 1993 SpringerVerlag.