Barker sequences and difference sets
type: Item
instance of: scholarly article
title: Barker sequences and difference sets
zbMATH Open document ID: 0777.05024
author: Shalom Eliahou
author: Michel Kervaire
published in: L'Enseignement Mathématique. 2e Série
publication date: 1993-06-29T00:00:00Z
review text: Let \(A=(a_ 1,\dots,a_ v)\) be a binary sequence (i.e. \(a_ i=\mp 1)\) such that the sums \(\gamma=\sum^ v_{i=1}a_ ia_{i+j}\) \((j=1,\dots,v-1)\) reading the indices modulo \(v\) do not depend on \(j\). Associating to \(A\) the set \(D=\{i\mid a_ i=-1\}\) we have a difference set on \(\mathbb{Z}_ v\), obtaining a bijection between such sequences \(A\) and cyclic difference sets. If \(|\gamma|\leq 1\), the sequence \(A\) is called a periodic Barker sequence. A nontrivial Barker sequence is a periodic Barker sequence, but it is not true that each periodic Barker sequence is a Barker sequence. On this line, and widely using the multiplier theorems (also as tools to see a difference set as union of orbits of a group of multipliers) the authors give a interesting survey (both on Barker sequences and difference sets) with many new proofs and results, mainly on existence questions. The case \(\gamma=0\) is equivalent to circular Hadamard sequences, but the authors prefer to discuss Turyn results on the possible values of \(v=4N^ 2\), \(N\geq 55\). For \(\gamma=1\) many nonexistence results are given in a table. The case \(\gamma=-1\) is equivalent to the study of Hadamard difference sets of order \(v=4n-1\) and an explicit complete classification up to \(n=100\) is given, of course using previous discussions and machine enumerations. Also usual (aperiodic) Barker sequence are studied: if \(v\) is an even integer having a prime factor \(p\equiv 3\pmod 4\), there is no Barker sequence of length \(v\), and so there is no nontrivial Barker sequence of even length less than 1,898,884.
Mathematics Subject Classification ID: 05B10
Mathematics Subject Classification ID: 05B20
Mathematics Subject Classification ID: 11B75
zbMATH DE Number: 221268
zbMATH Keywords: difference set
zbMATH Keywords: periodic Barker sequence
zbMATH Keywords: multiplier theorems
zbMATH Keywords: Hadamard sequences
zbMATH Keywords: Hadamard difference sets
reviewed by: Giovanni Ferrero
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 221268