Nested Hadamard difference sets
type: Item
instance of: scholarly article
title: Nested Hadamard difference sets
zbMATH Open document ID: 0881.05026
DOI: 10.1016/S0378-3758(96)00162-0
author: James A. Davis
author: Jonathan Jedwab
published in: Journal of Statistical Planning and Inference
publication date: 1998-01-22T00:00:00Z
review text: An Hadamard difference set (HDS) is a \((4n^2, 2n^2- n, n^2- n)\)-difference set. Hadamard difference sets on direct products of cyclic groups of prime-power orders are of particular interest due to their applications in signal processing, and have recently been studied by several authors. The main focus of the article is on Hadamard difference sets on abelian groups of the form \(H\times{\mathcal Z}^2_{p^\alpha}\), \(H\) a direct product of cyclic 2-groups, \(p\) a prime, and, in particular, the question of nonexistence of Hadamard difference sets for some of these groups. Using the correspondence between HDSs and perfect binary arrays together with the idea of a contraction by a subgroup, the authors were able to prove that an HDS on \(H\times{\mathcal Z}^2_{p^\alpha}\) implies the existence of an HDS for all \(H\times{\mathcal Z}^2_{p^k}\), \(1\leq k\leq\alpha\), provided the prime \(p\) is semiprimitive mod the exponent of \(H\). It follows that in such a case the nonexistence of an HDS on \(H\times{\mathcal Z}^2_p\) implies the nonexistence of HDSs for all \(H\times{\mathcal Z}^2_{p^k}\), \(k\geq 1\). This allowed them to show the nonexistence of HDSs for the groups \(H\times{\mathcal Z}^2_{7^k}\) and \(H\times{\mathcal Z}^2_{11^k}\), where \(H\) is any group of order 64 and exponent \(\leq 8\) and \(k\geq 1\).
Mathematics Subject Classification ID: 05B10
Mathematics Subject Classification ID: 62K05
zbMATH DE Number: 1057361
zbMATH Keywords: Hadamard difference set
zbMATH Keywords: perfect binary arrays
reviewed by: Robert Jajcay
MaRDI profile type: Publication
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full work available at URL: s0378-3758(96)00162-0
OpenAlex ID: W2093209961
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description: scientific article; zbMATH DE number 1057361