Quadratic double circulant codes over fields
type: Item
instance of: scholarly article
title: Quadratic double circulant codes over fields
zbMATH Open document ID: 1022.94018
published in: Journal of Combinatorial Theory. Series A
publication date: 2002-07-04T00:00:00Z
review text: The author generalizes and uniformizes some constructions of double circulant codes using quadratic residues. The general construction leads to what he calls QDC (quadratic double circulant) codes. Depending on the characteristic of the field over which the codes are defined, the construction contains and generalizes results by \textit{M. Karlin} [IEEE Trans. Inf. Theory 15, 81-92 (1969; Zbl 0167.18104)] for characteristic 2, \textit{G. F. M. Beenker} [ibid. 30, 403-405 (1984; Zbl 0541.94018)] and \textit{E. Dawson} [Ars Comb. 19 A, 303-308 (1985; Zbl 0558.94011)] for characteristic 3, \textit{J. Leon}, \textit{V. Pless} and \textit{N. Sloane} [J. Comb. Theory, Ser. A 32, 178-194 (1982; Zbl 0483.94018)] for characteristic 5, and improves results of \textit{V. Pless} and \textit{V. Tonchev} [IEEE Trans. Inf. Theory 33, 723-727 (1987; Zbl 0658.94016)] in the sense that the minimum distance of the newly constructed codes is higher given the other parameters. It is a pity that the author gives no general results on the minimum weight for the constructed codes. Instead of this he computed the minimum weight for low characteristics and moderate choices of other parameters, using the magma package. Nevertheless he improves on several of the results from the tables of \textit{A. Brouwer} [Handbook of coding theory (Elsevier; Amsterdam) (1998; Zbl 0936.94015)]. Some of the new results are even optimal according to these tables. Furthermore some results are given on automorphism groups of the constructed codes. The paper is a good basis for further research on QDC codes.
Mathematics Subject Classification ID: 94B05
Mathematics Subject Classification ID: 94B65
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 11T71
zbMATH DE Number: 1763795
zbMATH Keywords: quadratic double circulant codes
zbMATH Keywords: quadratic residues
zbMATH Keywords: automorphism groups
author: Philippe Gaborit
reviewed by: Herman J. Tiersma
describes a project that uses: Magma
describes a project that uses: Code Tables
MaRDI profile type: Publication
full work available at URL: jcta.2001.3198
OpenAlex ID: W2075359144
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DOI: 10.1006/JCTA.2001.3198
description: scientific article; zbMATH DE number 1763795