On Xia's construction of Hadamard difference sets
type: Item
instance of: scholarly article
title: On Xia's construction of Hadamard difference sets
zbMATH Open document ID: 0844.05020
author: Qing Xiang
author: Yu Qing Chen
published in: Finite Fields and their Applications
publication date: 1996-05-08T00:00:00Z
review text: An Hadamard difference set is a difference set with parameters of the form \(v= 4u^2\), \(k= 2u^2\pm u\) and \(\lambda= u^2\pm u\). In a major paper, \textit{M. Y. Xia} [Some infinite classes of special Williamson matrices and difference sets, J. Comb. Theory, Ser. 61, No. 2, 230-242 (1992; Zbl 0772.05022)] proved the following existence result: Let \(p\) be a prime congruent to 3 modulo 4. Then there exists an abelian Hadamard difference set in the direct product of \(H\) with the elementary abelian group of order \(p^4\), where \(H\) is either of the two groups of order 4. However, Xia's proof used very complicated calculations involving cyclotomic classes of high order and was therefore hard to understand. The paper under review contains a character theoretic, much more transparent proof of this important result. It also points out an important connection to projective three-weight codes. This has by now led to similar constructions for some primes congruent to 1 modulo 4, i.e. \(p= 5, 13\) and 17; see the manuscripts by \textit{M. van Eupen} and \textit{V. D. Tonchev} [The existence of a reversible Hadamard difference set in \(Z_2\times Z_2\times Z_{5^4}\) and an infinite class of Williamson matrices] and \textit{R. M. Wilson} and \textit{Q. Xiang} [Constructions of Hadamard difference sets].
Mathematics Subject Classification ID: 05B10
zbMATH DE Number: 851863
zbMATH Keywords: Hadamard difference set
reviewed by: Dieter Jungnickel
MaRDI profile type: Publication
OpenAlex ID: W1977213083
DOI: 10.1006/FFTA.1996.0006
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description: scientific article; zbMATH DE number 851863