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Automorphisms of higher‐dimensional Hadamard matrices
Author(s) -
de Launey Warwick,
Stafford Richard M.
Publication year - 2008
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.20200
Subject(s) - mathematics , automorphism , hadamard transform , complex hadamard matrix , combinatorics , order (exchange) , hadamard product , equivalence (formal languages) , hadamard's maximal determinant problem , hadamard matrix , group (periodic table) , matrix (chemical analysis) , discrete mathematics , mathematical analysis , physics , materials science , finance , quantum mechanics , economics , composite material
This article derives from first principles a definition of equivalence for higher‐dimensional Hadamard matrices and thereby a definition of the automorphism group for higher‐dimensional Hadamard matrices. Our procedure is quite general and could be applied to other kinds of designs for which there are no established definitions for equivalence or automorphism. Given a two‐dimensional Hadamard matrix H of order ν, there is a Product Construction which gives an order ν proper n ‐dimensional Hadamard matrix P (n) ( H ). We apply our ideas to the matrices P (n) ( H ). We prove that there is a constant c > 1 such that any Hadamard matrix H of order ν > 2 gives rise via the Product Construction to c ν inequivalent proper three‐dimensional Hadamard matrices of order ν. This corrects an erroneous assertion made in the literature that ”P (n) ( H ) is equivalent to “P (n) (H′) whenever H is equivalent to H′.” We also show how the automorphism group of P (n) ( H ) depends on the structure of the automorphism group of H . As an application of the above ideas, we determine the automorphism group of P (n) (H k ) when H k is a Sylvester Hadamard matrix of order 2 k . For ν = 4, we exhibit three distinct families of inequivalent Product Construction matrices P (n) ( H ) where H is equivalent to H 2 . These matrices each have large but non‐isomorphic automorphism groups. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 507–544, 2008