On optimal linear codes over \(\mathbb F_5\)
type: Item
instance of: scholarly article
title: On optimal linear codes over \(\mathbb F_5\)
zbMATH Open document ID: 1171.94011
author: Tatsuya Maruta
author: Maori Shinohara
author: Ayako Kikui
published in: Discrete Mathematics
publication date: 2009-06-17T00:00:00Z
review text: The authors solve some open cases of the main coding theory problem (which consists of finding the smallest value \(n_q(k,d)\) of \(n\) for which there exists a linear code of length \(n\) and fixed dimension \(k\) and minimal distance \(d\) over GF\((q)\), the finite field with \(q\) elements). In a table, lower and upper bounds for \(n_5(5,d)\) with \(d\leq 5^4\) are summarized (including some existence and non-existence results proved by the authors or obtained by computer search and including the results of an online table of M.\ Grassl). The table shows that many cases are still open. The geometric method used to construct or to prove the non-existence of some linear codes is to consider the points of a certain subset or multiset of a projective space over GF(5) as the columns of a generator matrix of the code; here the authors use as such subsets so-called minihypers constructed, for instance, from \(n\)-arcs or blocking sets.
reviewed by: Ralph-Hardo Schulz
Mathematics Subject Classification ID: 94B05
Mathematics Subject Classification ID: 94B27
Mathematics Subject Classification ID: 94B65
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 51E22
zbMATH DE Number: 5565667
zbMATH Keywords: optimal linear code
zbMATH Keywords: main coding theory problem
zbMATH Keywords: extension of a linear code
zbMATH Keywords: finite projective geometry
zbMATH Keywords: minihyper
zbMATH Keywords: arc
zbMATH Keywords: blocking set
zbMATH Keywords: bounds on codes
describes a project that uses: MinT
MaRDI profile type: Publication
full work available at URL: j.disc.2008.01.047
OpenAlex ID: W2064007968
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DOI: 10.1016/J.DISC.2008.01.047
description: scientific article; zbMATH DE number 5565667