A sign test for detecting the equivalence of Sylvester Hadamard matrices

In this paper we show that every matrix in the class of Sylvester Hadamard matrices of order 2k under H-equivalence can have full row and column sign spectrum, meaning that tabulating the numbers of sign interchanges along any row (or column) gives all integers 0,1,...,2k − 1 in some order. The construction and properties of Yates Hadamard matrices are presented and is established their equivalence with the Sylvester Hadamard matrices of the same order. Finally, is proved that every normalized Hadamard matrix has full column or row sign spectrum if and only if is H-equivalent to a Sylvester Hadamard matrix. This provides us with an efficient criterion identifying the equivalence of Sylvester Hadamard matrices.

[1]  C. Koukouvinos,et al.  Complementary, Base, and Turyn Sequences , 2006 .

[2]  C. Wilkins,et al.  Concise algorithm for conversion between normal and sequency ordered Hadamard matrices , 1974 .

[3]  C. Colbourn,et al.  Handbook of Combinatorial Designs , 2006 .

[4]  R. A. FISHER,et al.  The Design and Analysis of Factorial Experiments , 1938, Nature.

[5]  J. Sylvester.  LX. Thoughts on inverse orthogonal matrices, simultaneous signsuccessions, and tessellated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers , 1867 .

[6]  A. Street,et al.  Combinatorics: room squares, sum-free sets, Hadamard matrices , 1972 .

[7]  R. Plackett,et al.  THE DESIGN OF OPTIMUM MULTIFACTORIAL EXPERIMENTS , 1946 .

[8]  C. Colbourn,et al.  The CRC handbook of combinatorial designs , edited by Charles J. Colbourn and Jeffrey H. Dinitz. Pp. 784. $89.95. 1996. ISBN 0-8493-8948-8 (CRC). , 1997, The Mathematical Gazette.

[9]  V. Olshevsky,et al.  Ranks of Hadamard Matrices and Equivalence of Sylvester—Hadamard and Pseudo-Noise Matrices , 2007 .

[10]  R. Williams,et al.  Normal-to-Sequency-Ordered Hadamard Matrix conversion , 1976 .

[11]  Jennifer Seberry,et al.  Hadamard matrices, orthogonal designs and construction algorithms , 2003 .

[12]  Brendan D. McKay,et al.  Hadamard equivalence via graph isomorphism , 1979, Discret. Math..

[13]  K. Horadam.  Hadamard Matrices and Their Applications , 2006 .

[14]  W. D. Wallis,et al.  Designs 2002: Further Computational and Constructive Design Theory , 2012 .