Quadratic residue codes in their prime
type: Item
instance of: scholarly article
title: Quadratic residue codes in their prime
zbMATH Open document ID: 0752.94013
DOI: 10.1016/S0021-8693(05)80051-1
author: Harold N. Ward
published in: Journal of Algebra
publication date: 1992-09-27T00:00:00Z
review text: The standard methods for constructing quadratic residue codes require that the characteristic \(p\) of the base field \(GF(q)\) from which they are obtained be invertible in the alphabet ring. However, the spanning sets created do not show this, and they can be read into a field of characteristic \(p\). The resulting codes are invariant under a monomial action of \(GL(2,q)\), rather than one just of \(SL(2,q)\), as for the conventional codes. The invariant subspaces of the action can be produced; the lattice of these additional codes turns out to depend only on the exponent in \(q\) as a power of \(p\). This paper describes the codes and investigates their dimensions and minimum weights. One of them is an extremal type III code of length 28, and certain others are reminiscent of Reed-Muller codes.
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 63896
zbMATH Keywords: quadratic residue codes
zbMATH Keywords: monomial action
zbMATH Keywords: dimensions
zbMATH Keywords: minimum weights
zbMATH Keywords: extremal type III code
zbMATH Keywords: Reed-Muller codes
MaRDI profile type: Publication
cites work: New 5-designs
cites work: Q3688305
cites work: Q4039784
cites work: On generalized ReedMuller codes and their relatives
cites work: The submodule structure of certain Weyl modules for groups of type \(A_ n\)
cites work: The automorphism groups of Reed-Solomon codes
cites work: Über die Quadratwurzel-Schranke für quadratische-Rest-Codes. (On the square root bound for quadratic-residue codes)
cites work: Some results on cyclic codes which are invariant under the affine group and their applications
cites work: Generalized quadratic residue codes
cites work: Q3946078
cites work: On the minimum distance of cyclic codes
cites work: An upper bound for self-dual codes
cites work: On minimum weight codewords in QR codes
cites work: Quadratic residue codes and symplectic groups
cites work: Quadratic residue codes of length 27
description: scientific article; zbMATH DE number 63896