Some progress on \((v,4,1)\) difference families and optical orthogonal codes
type: Item
instance of: scholarly article
title: Some progress on \((v,4,1)\) difference families and optical orthogonal codes
zbMATH Open document ID: 1061.94068
author: R. Julian R. Abel
author: Marco Buratti
published in: Journal of Combinatorial Theory. Series A
publication date: 2004-08-06T00:00:00Z
review text: The authors consider collections \({\mathcal C}\) of \(k\)-subsets of a \(v\)-set such that any two sets intersect in at most one point. In the special case \(k=4\), such packings are the same objects as constant-weight-codes with weight \(4\) and minimum distance at least \(6\). If the code associated with the packing is cyclic, it is called an optical orthogonal code (OOC). Given \(v\) and \(k\), there is a trivial upper bound for the number of codewords in an OOC. If this bound is reached, the code is called optimal. Constructions of OOC's use modifications of the well-known difference techniques in design theory (cyclic difference families for designs with blocksize \(4\) are, for instance, special types of OOC's). Using difference methods, the authors construct new families of OOC's of maximum size where \(v\) is prime and satisfies some number theoretic conditions. These new examples are used to construct optimal OOC's of length \(5v\) (if \(v\equiv 5\mod 12\)) and \(7v\) (if \(v\equiv 7\mod 12\)). The latter examples are actually difference families. Moreover, it is shown that cyclic difference families with blocksize \(4\) exist for all \(v\leq 601\), except \(v=25\). Optimal OOC's exist for all \(v\leq 408\), except \(v=25\).
Mathematics Subject Classification ID: 94B25
Mathematics Subject Classification ID: 05B10
Mathematics Subject Classification ID: 05B40
zbMATH DE Number: 2081940
zbMATH Keywords: optical orthogonal code
zbMATH Keywords: difference family
zbMATH Keywords: difference triangle set
zbMATH Keywords: Skolem sequence
reviewed by: Alexander Pott
MaRDI profile type: Publication
full work available at URL: j.jcta.2004.01.003
OpenAlex ID: W2076396106
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DOI: 10.1016/J.JCTA.2004.01.003
description: scientific article; zbMATH DE number 2081940