On non-symmetric relative difference sets
type: Item
instance of: scholarly article
title: On non-symmetric relative difference sets
zbMATH Open document ID: 1176.05014
author: Yutaka Hiramine
published in: Hokkaido Mathematical Journal
publication date: 2008-10-20T00:00:00Z
review text: An \((m,u,k,\lambda)\)\textit{-relative difference set} in a group \(G\) of order \(mu\) relative to a subgroup \(U\) of order \(u\) is a \(k\)-subset \(D\) of \(G\) with the property that the multiset of non-identity products \(d_1d_2^{-1}\), \(d_1,d_2 \in D \), contains each element of \(G \setminus U \) \(\lambda\) times, and contains no elements of \(U\). The \((m,u,k,\lambda)\)-relative difference set \(D\) is called \textit{symmetric} if \(D^{-1} = \{ d^{-1} \; | \; d \in D \} \) is also an \((m,u,k,\lambda)\)-relative difference set for the same group/subgroup pair. If the index \(m\) of \(U\) in \(G\) is equal to the size \(k\) of \(D\) then \(D\) is called \textit{semiregular}. Given a relative difference set \(D\), the associated incidence structure \(dev(D)\) with the point set \(G\) and the block set \( \{ Dg \; | \; g \in G \} \) is known to be an \((m,u,k,\lambda)\)\textit{-divisible design}. In the main result of the paper, the author shows that a semiregular relative difference set \(D\) is symmetric if and only if the dual of \(dev(D)\) is also a divisible design. This is followed by a recursive construction of an infinite family of semiregular non-symmetric relative difference sets. The construction uses a modification of Davis' recursive construction [\textit{J. A. Davis}, ``A note on products of relative difference sets'', Des. Codes Cryptography 1, No.2, 117--119 (1991; Zbl 0754.05014)] started from a non-symmetric \((12,3,12,4)\)-relative difference set.
Mathematics Subject Classification ID: 05B10
Mathematics Subject Classification ID: 05B05
zbMATH DE Number: 5354265
zbMATH Keywords: relative difference set
zbMATH Keywords: subgroup
zbMATH Keywords: symmetric
zbMATH Keywords: divisible design
zbMATH Keywords: transversal design
reviewed by: Robert Jajcay
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1253539528
OpenAlex ID: W2078673868
DOI: 10.14492/HOKMJ/1253539528
Recommended article: Q4878688
Recommended article: Q3067983
Recommended article: New constructions of divisible designs
Recommended article: A note on new semi-regular divisible difference sets
Recommended article: New semiregular divisible difference sets
Recommended article: Q4934411
Recommended article: Q4271744
Recommended article: Q4437291
Recommended article: Construction of relative difference sets in \(p\)-groups
description: scientific article; zbMATH DE number 5354265