SupposeA 1,...,A s are (1, - 1) matrices of order m satisfying
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
if there exist a regular s-set of order m and a regulart-set of order n there exists a regulars-set of ordermn whent =sm
if there exist a regular s-set of order m and a regulart-set of order n there exists a regulars-set of ordermn when 2t = sm (m is odd)
if there exist a regulars-set of order m and a regulart-set of ordern there exists a regular 2s-set of ordermn whent = 2sm As applications, we prove that if there exist a regulars-set of order m there exists
an Hadamard matrices of order4hm whenever there exists an Hadamard matrix of order4h ands =2h
Williamson type matrices of ordernm whenever there exists Williamson type matrices of ordern and s = 2n
anOD(4mp;ms1,…,msu whenever anOD (4p;s1,…,su)exists and s = 2p
a complex Hadamard matrix of order 2cm whenever there exists a complex Hadamard matrix of order 2c ands = 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.
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Seberry, J., Zhang, X. Regular sets of matrices and applications. Graphs and Combinatorics 9, 185 (1993). https://doi.org/10.1007/BF02988305
- Type Matrice
- Prime Power
- Orthogonal Design
- Hadamard Matrice
- Hadamard Matrix