Automorphisms of higher-dimensional Hadamard matrices

✍ Scribed by Warwick de Launey; Richard M. Stafford

• Publisher: John Wiley and Sons
• Year: 2008
• Tongue: English
• Weight: 308 KB
• Volume: 16
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.20200

✦ Synopsis

Abstract

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