Hadamard matrices of composite orders

Document Type : Research Paper


1 School of Computing and Information Technology, University of Wollongong Australia, Wollongong, Australia

2 School of Mathematics and Statistics, Central China Normal University, Wuhan, China

3 College of Mathematics and Statistics, South-Central University for Nationalities, Wuhan, China


In this paper, we give a method for the constructions of Hadamard matrices of composite orders by using suitable $T$-matrices and known Hadamard matrices. We establish a formula for $T$-matrices and Hadamard matrices and discuss under what condition we can get $T$-matrices from the known Hadamard matrices.


Main Subjects

[1] Y. Q. Chen, On the existence of Abelian Hadamard difference sets and a new family of difference sets,
Finite Fields Appl., 3 (1997) 234–256.
[2] C. Koukouvinos, Base sequence of length m + p, m + p, m, m for p = 1 and 0 ≤ 0 ≤ 35, and p = 2t − 1,
m = 2t, 13 ≤ t ≤ 17, Personal communication.
[3] J. Seberry and M. Yamada, Hadamard matrices, sequences, and block designs in Contemporary Design
Theory, Wiley, New York, 1992 431–560.
[4] M. Xia, Some infinite classes of special Williamson matrices and difference sets, J. Comb. Theory, Ser. A,
61 (1992) 230–242.
[5] M. Xia and T. Xia, Hadamard matrices constructed from supplementary difference sets in the class H1 , J.
Comb. Des., 2 (1994) 325–339.
[6] M. Xia and T. Xia, A family of C-partitions and T -matrices, J. Comb. Des., 7 (1999) 269–281.
[7] M. Xia, T. Xia and J. Seberry, A new method for Constructing Williamson matrices, Des. Codes Cryptog-
raphy, 35 (2005) 191–209.
[8] T. Xia, G. Zuo and M. Xia, 2 Families of negacyclic matrices, Far East Journal of Mathematical Science,
129 (2021) 131–146.
[9] T. Xia, J. Seberry and M. Xia, Some new constructions of orthogonal designs, Australas. J. Comb., 55
(2013) 121–130.
[10] G. Zuo and T. Xia, A special class of T -matrices, Des. Codes Cryptography, 54 (2010) 21–28.
[11] G. Zuo, M. Xia and T. Xia, Constructions of composite T -matrices, Linear Algebra Appl., 438 (2013)
[12] H. Gholamiangonabadi, Amicable T -matrices and Applications, M. Sc. Thesis, University of Lethbridge,
Alberta, Canada, 2012