Semi-regular Sets of Matrices and Applications

The concept of semi-regular sets of matrices was introducted by J. Seberry in " A new construction for Williamson-type matrices ", Graphs and Combinatorics, 2(1986), 81-87. A regular s-set of matrices of order m. was 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. X. M. Zhang studied the product of these matrices and their applications in " Regular sets of matrices and applications " (to appear). In this paper, we prove that (i) if there exist a semi-regular s-set of order m and a semi-regular t-set of order n then there exists a semi-regular s-set of order mn when t = sm, (ii) if there exist Williamson type matrices of order n and a semiregular s(= 2n)-set of matrices of order m then there exist Williamson type matrices of order nm, (iii) if there exists a complex Hadamard matrix of order 2c and a semi-regular s(= 2c)-set of matrices of order m th..