Description of minimum weight codewords of cyclic codes by algebraic systems - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles Finite Fields and Their Applications Year : 1996

Description of minimum weight codewords of cyclic codes by algebraic systems

Daniel Augot
  • Function : Author
  • PersonId : 833459

Abstract

We consider cyclic codes of length n over ‫ކGF(q), n being prime to q. Fo such a cyclic code C, we describe a system of algebraic equations, denoted by SC(w), where w is a positive integer. The system is constructed from Newton's identities, which are satisfied by the elementary symmetric functions and the (generalized) power sum symmetric functions of the locators of codewords of weight w. The main result is that, in a certain sense, the algebraic solutions of SC(w) are in one-to-one correspondence with all the codewords of C having weight lower than w. In the particular case where w is the minimum distance of C, all minimum weight codewords are described by SC(w). Because the system SC (w) is very large, with many indeterminates, no great insight can be directly obtained, and specific tools are required in order to manipulate the algebraic systems. For this purpose, the theory of Groebner bases can be used. A Groebner basis of SC (w) gives information about the minimum weight codewords.
Fichier principal
Vignette du fichier
ffa.pdf (100.48 Ko) Télécharger le fichier
Origin : Publisher files allowed on an open archive
Loading...

Dates and versions

hal-00723500 , version 1 (10-08-2012)

Identifiers

Cite

Daniel Augot. Description of minimum weight codewords of cyclic codes by algebraic systems. Finite Fields and Their Applications, 1996, 2, pp.138-152. ⟨10.1006/ffta.1996.0009⟩. ⟨hal-00723500⟩

Collections

INRIA INRIA2
105 View
276 Download

Altmetric

Share

Gmail Facebook X LinkedIn More