### Abstract

Suppose A_{1},..., A_{s} are (1, - 1) matrices of order m satisfying {Mathematical expression} {Mathematical expression} {Mathematical expression} {Mathematical expression} Call A_{1},..., 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;ms_{1},...,ms_{u} whenever an OD (4p;s_{1},...,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.

Original language | English |
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Pages (from-to) | 185-195 |

Number of pages | 11 |

Journal | Graphs and Combinatorics |

Volume | 9 |

Issue number | 2-4 |

DOIs | |

Publication status | Published - Jun 1993 |

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## Cite this

*Graphs and Combinatorics*,

*9*(2-4), 185-195. https://doi.org/10.1007/BF02988305