Codes and finite geometries

Abstract : We explore the connections between finite geometry and algebraic coding theory, giving a rather full account of the Reed-Muller and generalized Reed-Muller codes. Some of the results and many of the proofs are new but this is largely an expository effort that relies heavily on the work of Delsarte et al. and of Charpin. The necessary geometric background is sketched before we begin the discussion of the Reed-Muller codes and their p-ary analogues. We prove all the classical results concerning these codes and include a discussion of the group-algebra approach and prove Berman's theorem characterizing the codes as powers of the radical. Included also is a discussion of the characterization of affine-invariant cyclic codes given by Kasami, Lin and Peterson and its generalization by Delsarte. our theme throughout this work is the relationship between these codes and the codes coming from both affine and projective geometries. The final section develops the theory in the more difficult case in which the field is not of prime order, here must look at subfield subcodes - which complicates the connection with the geometric codes, which are codesover the prime subfield of the field of the geometry.
Type de document :
Rapport
[Research Report] RR-2027, INRIA. 1993
Liste complète des métadonnées

https://hal.inria.fr/inria-00074644
Contributeur : Rapport de Recherche Inria <>
Soumis le : mercredi 24 mai 2006 - 15:58:53
Dernière modification le : vendredi 25 mai 2018 - 12:02:03
Document(s) archivé(s) le : mardi 12 avril 2011 - 17:58:16

Fichiers

Identifiants

  • HAL Id : inria-00074644, version 1

Collections

Citation

Edwar F. Jr Assmus, Jennifer D. Key. Codes and finite geometries. [Research Report] RR-2027, INRIA. 1993. 〈inria-00074644〉

Partager

Métriques

Consultations de la notice

86

Téléchargements de fichiers

70