The weight hierarchies of the projective codes from nondegenerate quadrics
type: Item
instance of: scholarly article
title: The weight hierarchies of the projective codes from nondegenerate quadrics
zbMATH Open document ID: 0802.94014
DOI: 10.1007/BF01388457
published in: Designs, Codes and Cryptography
publication date: 1994-12-15T00:00:00Z
review text: Let \(C\) be a linear code over the finite field \(F_ q\). \(C\) is called a projective code if the columns of a generator matrix of \(C\) are not proportional. Each column can be regarded as a point in the \((k-1)\)- dimensional projective space over \(F_ q\). Projective codes whose generator matrix columns satisfy a nondegenerate quadric are considered here. For a linear subcode \(D\) of \(C\) let \(\chi(D)\) denote the support of \(D\) defined by \(\chi (D)= \{i\mid x_ i\neq 0\) for some \((x_ 1, x_ 2, \dots, x_ n)\in D\}\). The \(r\)-th generalized Hamming weight of \(C\), denoted by \(d_ r(C)\), is \[ d_ r(C)= \min\{ |\chi(D)| \mid D \text{ is an \(r\)-dimensional subcode of } C\}. \] The weight hierarchy of \(C\) is \(\{d_ 1 (C), d_ 2(C), \dots, d_ r(C)\}\). The code \(C\) satisfies the chain condition if there exists an \(r\)-dimensional subcode \(D_ r\) of \(C\) such that \(|\chi (D_ r)|= d_ r(C)\), \(r=1,2,\dots, k\) and \(D_ 1 \subseteq D_ 2 \subseteq \cdots \subseteq D_ k\). The weight hierarchies of the projective codes from nondegenerate quadrics in projective spaces are calculated here. It is shown the codes satisfy the chain condition.
Mathematics Subject Classification ID: 94B05
Mathematics Subject Classification ID: 94B25
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 51E05
zbMATH DE Number: 597836
zbMATH Keywords: linear codes
zbMATH Keywords: projective geometries
zbMATH Keywords: projective codes
zbMATH Keywords: nondegenerate quadrics
author: Zhe-Xian Wan
reviewed by: Ian Blake
MaRDI profile type: Publication
cites work: Generalized Hamming weights for linear codes
cites work: The weight distribution of irreducible cyclic codes with block lengths \(n_1 ((q^\ell-1)/N)\)
cites work: On the generalized Hamming weights of product codes
cites work: Generalized Hamming weights of linear codes
cites work: Q4310042
cites work: Q5804609
cites work: Codes projectifs à deux ou trois poids associes aux hyperquadriques d'une géométrie finie
full work available at URL: https://doi.org/10.1007/bf01388457
OpenAlex ID: W1978435645
description: scientific article; zbMATH DE number 597836