Regular sets of matrices and applications

Suppose A₁,..., A_s are (1, - 1) matrices of order m satisfying {Mathematical expression} {Mathematical expression} {Mathematical expression} {Mathematical expression} Call A₁,..., A_s, a regular s- set of matrices of order m if Eq. 1-3 are satisfied and a regular s-set 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), 355-377. In this paper, we prove that (i) if there exist a regular s-set of order m and a regular t-set of order n there exists a regular s-set of order mn when t =sm (ii) if there exist a regular s-set of order m and a regular t-set of order n there exists a regular s-set of order mn when 2t = sm (m is odd) (iii) if there exist a regular s-set of order m and a regular t-set of order n there exists a regular 2s-set of order mn when t = 2sm As applications, we prove that if there exist a regular s-set 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,...,ms_u whenever an OD (4p;s₁,...,s_u)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.