Reed-Muller-type codes over the Segre variety
type: Item
instance of: scholarly article
title: Reed-Muller-type codes over the Segre variety
zbMATH Open document ID: 1020.94029
published in: Finite Fields and their Applications
publication date: 2003-04-02T00:00:00Z
review text: The image of the Segre embedding \(\mathbb P^n_k \times \mathbb P^m_k\rightarrow \mathbb P^N_k\), where \(N=(n+1)(m+1)-1\), is the Segre variety \(\Sigma_{n,m}\). If \(K\) is a finite field, the authors consider the evaluation code \(C(d)\) obtained by evaluating homogeneous polynomials of degree \(d\) on (representatives of) the \(K\)-rational points of \(\Sigma_{n,m}\). The Hilbert function of \(\Sigma_{n,m}\) is used to compute the dimension of \(C(d)\). The authors show that the minimum distance of \(C(d)\) is \(\delta_{\mathbb P^n}(d)\delta_{\mathbb P^m}(d)\), where \(\delta_{\mathbb P^l}(d)\) is the minimum distance of the projective Reed-Muller code obtained by evaluating homogeneous polynomials of degree \(d\) on the \(K\)-rational points of \(\mathbb P^l\) (cf. \textit{G. Lachaud} [Projective Reed-Muller codes. In: Coding theory and applications, Proc. 2nd Int. Colloq., Paris/Fr. 1986, Lect. Notes Comput. Sci. 311, 125--129 (1988; Zbl 0651.94008)] and \textit{A. B. Sørensen} [IEEE Trans. Inf. Theory 37, 1567--1576 (1991; Zbl 0741.94016)]). An example is given with \(n=m=1\), \(d=2\) and \(K={\mathbb F}_4\).
Mathematics Subject Classification ID: 94B27
Mathematics Subject Classification ID: 14G50
Mathematics Subject Classification ID: 14G15
zbMATH DE Number: 1891463
zbMATH Keywords: projective Reed-Muller code
zbMATH Keywords: Segre variety
zbMATH Keywords: minimum distance
author: Manuel González Sarabia
author: Carlos Rentería Márquez
author: Horacio Tapia-Recillas
reviewed by: Robert F. Lax
MaRDI profile type: Publication
full work available at URL: ffta.2002.0360
OpenAlex ID: W2026545856
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DOI: 10.1006/FFTA.2002.0360
description: scientific article; zbMATH DE number 1891463