Linear spaces of quadrics and new good codes
type: Item
instance of: scholarly article
title: Linear spaces of quadrics and new good codes
zbMATH Open document ID: 0929.51007
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2000-01-23T00:00:00Z
review text: Let \(V\) be a vector space over the finite field \(F_q\) and \(Q\) the space of all quadratic forms on \(V\). From a subspace \(G\) of \(Q\) a linear code can be constructed: each codeword is obtained by evaluating a quadratic form of \(G\) in fixed representatives of all points in the projective space \(PV\). In his PhD thesis [`Codes: their parameters and geometry' (1997; Zbl 0866.94024)], \textit{M. A. de Boer} constructs a family of subspaces of \(Q\) that result in codes with high minimum distance. Moreover some parameters of these codes are computed. Conjecture 4.2.10 of the thesis deals with the number of nonzero weights of some of the constructed codes. In the current paper Brouwer proves Conjecture 4.2.10 of the thesis by de Boer and presents a shorter proof for the results on the parameters of the codes. Brouwer ends with a discussion on how to improve on the codes, thus obtaining codes that have higher minimum distance than all previously known codes.
Mathematics Subject Classification ID: 51E22
Mathematics Subject Classification ID: 94B27
zbMATH DE Number: 1266804
zbMATH Keywords: quadratic forms
zbMATH Keywords: linear codes
author: Andries E. Brouwer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1266804