Divisibility of codes meeting the Griesmer bound
type: Item
instance of: scholarly article
title: Divisibility of codes meeting the Griesmer bound
zbMATH Open document ID: 0913.94026
author: Harold N. Ward
published in: Journal of Combinatorial Theory. Series A
publication date: 1999-05-31T00:00:00Z
review text: The Griesmer bound is a well-known bound on the length of a linear \([n,k,d]\) code, and codes which meet this bound are optimal. A divisor of a linear code is an integer dividing the weights of all its codewords, and a code is called divisible if it has a divisor larger than 1. Optimal codes are often divisible. This paper proves the following result: Theorem 1. Let \(C\) be an \([n,k,d]\) code over \(\text{GF}(p)\), \(p\) a prime, meeting the Griesmer bound. If \(p^e | d\), then \(p^e\) is a divisor of \(C\). Note that for large enough \(d\), codes exist which meet the Griesmer bound. A number of examples of divisible codes are given.
reviewed by: T. Aaron Gulliver
Mathematics Subject Classification ID: 94B65
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 1212912
zbMATH Keywords: code divisor
zbMATH Keywords: Griesmer bound
MaRDI profile type: Publication
OpenAlex ID: W2080231105
cites work: New 5-designs
cites work: A note on the Griesmer bound (Corresp.)
cites work: Q5187201
cites work: Optimal linear codes of dimension 4 over F/sub 5/
cites work: The correspondence between projective codes and 2-weight codes
cites work: Minimum block length of a linear q-ary code with specified dimension and code distance
cites work: Q3698695
cites work: A comment on the weight structure of generator matrices of linear codes
cites work: Minimum possible block length of a linear binary code for some distances
cites work: The nonexistence of ternary \([50,5,32]\) codes
cites work: Linear codes and the existence of a reversible Hadamard difference set in \(\mathbb{Z}_ 2\times \mathbb{Z}_ 2\times \mathbb{Z}_ 5^ 4\)
cites work: Q4201554
cites work: Caps and codes
cites work: Q4011770
cites work: Q3838747
cites work: Optimal ternary linear codes
cites work: Q3838750
cites work: Q5665102
cites work: Divisible codes
cites work: Weight polarization and divisibility
cites work: Characters and the equivalence of codes
cites work: Constructions of Hadamard difference sets
DOI: 10.1006/JCTA.1997.2864
Recommended article: A bound for divisible codes
Recommended article: The divisible code bound revisited
Recommended article: On codes meeting the Griesmer bound.
Recommended article: Further classifications of codes meeting the Griesmer bound (Corresp.)
Recommended article: On the Lengths of Divisible Codes
Recommended article: Q3791083
Recommended article: Characterization of two classes of codes that attain the Griesmer bound
Recommended article: New constructions of codes meeting the Griesmer bound
Recommended article: A Griesmer bound for codes over finite quasi-Frobenius rings
Recommended article: A new class of nonbinary codes meeting the Griesmer bound
description: scientific article; zbMATH DE number 1212912