Abstract
SupposeA _{1},...,A _{ s } are (1,  1) matrices of order m satisfying
Call A_{1},…,A _{ s },a regular s set of matrices of order m if Eq. 13 are satisfied and a regular sset 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), 355377. In this paper, we prove that

(i)
if there exist a regular sset of order m and a regulartset of order n there exists a regularsset of ordermn whent =sm

(ii)
if there exist a regular sset of order m and a regulartset of order n there exists a regularsset of ordermn when 2t = sm (m is odd)

(iii)
if there exist a regularsset of order m and a regulartset of ordern there exists a regular 2sset of ordermn whent = 2sm As applications, we prove that if there exist a regularsset of order m there exists

(iv)
an Hadamard matrices of order4hm whenever there exists an Hadamard matrix of order4h ands =2h

(v)
Williamson type matrices of ordernm whenever there exists Williamson type matrices of ordern and s = 2n

(vi)
anOD(4mp;ms_{1},…,ms_{u} whenever anOD (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 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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References
 1.
Seberry, J., Whiteman, A.L. New Hadaraard matrices and conference matrices obtained via Mathon’s construction. Graphs Comb.4, 355–377 (1988)
 2.
Seberry, J., Yamada, M. Hadamard matrices, sequences and block designs, in: J. Dinitz, D. Stinson (eds) Contemporary Design Theory. WileyInterscience Series in Discrete Mathematics, New York: John Wiley 431–560 (1992)
 3.
Baumert, L.D. Hall, J.M. Hadamard matrices of Williamson type. Math. Comp.,19, 442–447 (1965)
 4.
Yamamoto, K. On a generalized Williamson equation. Colloq. Math. Soc. Janos Bolyai,37, 839–850 (1981)
 5.
Yamamoto, K., Yamada, M. Williamson matrices of Turyn’s type and Gauss sums. J. Math. Soc. Japan,37, 703–717 (1985)
 6.
Seberry, J. A new construction for Williamsontype matrices. Graphs Comb.,2, 81–87 (1981)
 7.
Wallis, J.S. Construction of Williamson type matrices. Linear Multilinear Algebra,3, 197–207 (1975)
 8.
Seberry, J. Some matrices of Williamsontype. Utilitas Math.,4, 147–154 (1973)
 9.
Whiteman, A.L. An infinite family of Hadamard matrices of Williamson type. J. Comb. Theory, Ser. A14, 334–340 (1973)
 10.
Whiteman, A.L. Hadamard matrices of Williamson. J. Austral. Math. Soc.,21, 481–486 (1976)
 11.
Miyamoto, M. A construction for Hadamard matrices. Comb. Theory. Ser. A,57, 86–108 (1991)
 12.
Yamada, M. On the Williamson matrices of Turyn’s type and type j. Comment. Math. Univ. St. Pauli,31, 71–73 (1982)
 13.
Seberry, J., Yamada, M. On the products of Hadamard matrices, Williamson matrices and other orthogonal matrices usingM structures. JCMCC,7, 97–137 (1990)
 14.
Wallis, W.D., Street, A.P., Wallis, J.S. Combinatorics: Room Squares, Sumfree Sets, Hadamard Matrices. vol. 292 of Lecture Notes in Mathematics. Berlin (Heidelberg New York: SpringerVerlag) 1972
 15.
Kharagani, H., Seberry, J. Regular complex Hadamard matrices. Congr. Numerantium,24, 149–151 (1990)
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Correspondence to Jennifer Seberry or XianMo Zhang.
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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
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Keywords
 Type Matrice
 Prime Power
 Orthogonal Design
 Hadamard Matrice
 Hadamard Matrix