Codes with given distances
type: Item
instance of: scholarly article
title: Codes with given distances
zbMATH Open document ID: 0629.94014
DOI: 10.1007/BF01788526
author: B. George
published in: Graphs and Combinatorics
publication date: 1987-01-01T00:00:00Z
review text: Let \(F\) be the field with two elements and \(C\) be a code of length \(n\) over \(F\), i.e. a subset of the \(n\)-dimensional vector space over \(F\). Let \(D(C)=\{d(x,y),\ x,y\in C,\ x\ne y\}\), where \(d(x,y)\) denotes the Hamming distance between \(x\) and \(y\). This paper concerns several problems about codes for which \(D(C)\) does not contain some prescribed values. One of the main results says that if \(C\) is linear of length \(4t\) and dimension at least \(2t\), then \(2t-1\) or \(2t\) belongs to \(D(C)\). Using this result the authors give a short proof for Gleason's theorem; if \(C\) is a linear code of length \(n\) which is self-dual and \(4\mid d\) for every \(d\in D(C)\), then \(8\mid n\). Another consequence of the main result is that if \(F_1,\ldots, F_{2t-1}\) are subsets of \(X\), \(| X| =4t\), then there exists a \(G\le X\) of size \(2t\) such that \(| G\cap F_i|\) is even for all \(i\), \(1\le i\le 2t-1\). For the proof of the main results a lemma introducing the binormal forms of binary matrices is needed.
Mathematics Subject Classification ID: 94B05
Mathematics Subject Classification ID: 94B25
Mathematics Subject Classification ID: 05C15
zbMATH DE Number: 4025329
zbMATH Keywords: Hamming distance
zbMATH Keywords: Gleason's theorem
zbMATH Keywords: binormal forms of binary matrices
reviewed by: Attila Pethő
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01788526
OpenAlex ID: W1973652235
description: scientific article; zbMATH DE number 4025329