Regular sets of matrices and applications

Suppose A 1 ; ; A s are (1;?1) matrices of order m satisfying (4) Call A 1 ; ;A s a regular s-set of matrices of order m if (1), (2), (3) are satissed and a regular s-set of regular matrices if (4) is also satissed, these matrices were rst 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 then 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 then 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 then 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 matrix 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) 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 and complex Hadamard matrices. In particular, we show that if both p 1 (mod 4) and 2p + 1 are prime powers then there exist Williamson type matrices of order 1 2 (p + 1)(2p + 1) 2 .