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Regular sets of matrices and applications

Journal Article


Abstract


  • Suppose A1,..., As are (1, - 1) matrices of order m satisfying {Mathematical expression} {Mathematical expression} {Mathematical expression} {Mathematical expression} Call A1,..., As, 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,...,msu whenever an OD (4p;s1,...,su)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. © 1993 Springer-Verlag.

Publication Date


  • 1993

Citation


  • Seberry, J., & Zhang, X. M. (1993). Regular sets of matrices and applications. Graphs and Combinatorics, 9(2-4), 185-195. doi:10.1007/BF02988305

Scopus Eid


  • 2-s2.0-77951516390

Web Of Science Accession Number


Start Page


  • 185

End Page


  • 195

Volume


  • 9

Issue


  • 2-4

Abstract


  • Suppose A1,..., As are (1, - 1) matrices of order m satisfying {Mathematical expression} {Mathematical expression} {Mathematical expression} {Mathematical expression} Call A1,..., As, 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,...,msu whenever an OD (4p;s1,...,su)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. © 1993 Springer-Verlag.

Publication Date


  • 1993

Citation


  • Seberry, J., & Zhang, X. M. (1993). Regular sets of matrices and applications. Graphs and Combinatorics, 9(2-4), 185-195. doi:10.1007/BF02988305

Scopus Eid


  • 2-s2.0-77951516390

Web Of Science Accession Number


Start Page


  • 185

End Page


  • 195

Volume


  • 9

Issue


  • 2-4