The weight enumerator of linear codes over \(\mathrm{GF}(q^m)\) having generator matrix over \(\mathrm{GF}(q)\)
type: Item
instance of: scholarly article
title: The weight enumerator of linear codes over \(\mathrm{GF}(q^m)\) having generator matrix over \(\mathrm{GF}(q)\)
zbMATH Open document ID: 0934.94021
DOI: 10.1023/A:1008330510348
published in: Designs, Codes and Cryptography
publication date: 2000-04-24T00:00:00Z
review text: Let \(C\) be an \([n,k]\)-code over \(\mathrm{GF}(q)\) with generator matrix \(G\). Let \(C^{(m)}\) denote the code over \(\mathrm{GF}(q^m)\) generated by the same matrix. \textit{T. Helleseth}, \textit{T. Kløve} and \textit{J. Mykkeltveit} [Discrete Math. 18, 179--211 (1977; Zbl 0357.94007)] gave an expression for the (Hamming) weight enumerator of \(C^{(m)}\) involving the number of \((k-j)\)-dimensional subspaces of \(\mathrm{GF}F(q)^k\) that contain exactly \(n-i\) columns of \(G\). Here, an alternative expression for this weight enumerator is given by using the \(\lambda\)-ply weight enumerator introduced by the author in a previous paper [Hokkaido Math. J. 25, 651--656 (1996; Zbl 0868.05006)]. As an application, the author determines the support weight enumerator (giving the number of \(r\)-dimensional subcodes of \(C\) with support weight \(i\)) for MDS codes.
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 1275106
zbMATH Keywords: Hamming weight enumerator
zbMATH Keywords: \(\lambda\)-ply weight enumerator
zbMATH Keywords: support weight enumerator
zbMATH Keywords: linear codes
zbMATH Keywords: MDS codes
author: Keisuke Shiromoto
reviewed by: Robert F. Lax
MaRDI profile type: Publication
full work available at URL: a:1008330510348
OpenAlex ID: W2063487149
description: scientific article; zbMATH DE number 1275106