940 articles – 1212 references  [version française]
 HAL: hal-00719506, version 1
 Available versions: v1 (2012-07-20) v2 (2013-05-13)
 Self-dual skew codes and factorization of skew polynomials
 (2012)
 In previous work the authors generalized cyclic codes to the noncommutative polynomial setting and used this approach to construct new self-dual codes over F4. According to this previous result, such a self-dual code must be $\theta$-constacyclic, i.e. the generator polynomial is a right divisor of some noncommutative polynomial $X^n-a$. The first result of the paper is that such a self-dual code must be $\theta$-cyclic or $\theta$-negacyclic, i.e. $a=\pm 1$. For codes of length $2^s$ the noncommutative polynomial approach produced surprisingly poor results. We give an explanation of the length $2^s$ phenomena by showing that in this case the generating skew polynomial has some unique factorization properties. We also construct self-dual skew codes using least common left multiples of noncommutative polynomials and use this to obtain a new $[78,39,19]_4$ self-dual code.
 1: Institut de Recherche Mathématique de Rennes (IRMAR) CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
 Research team: Géométrie algébrique réelle
 Subject : Mathematics/Rings and Algebras
 Keyword(s): codes correcteurs d'erreurs – corps finis – polynômes non commutatifs
Attached file list to this document:
 PDF
 soumission_HAL.pdf(181.2 KB)
 hal-00719506, version 1 http://hal.archives-ouvertes.fr/hal-00719506 oai:hal.archives-ouvertes.fr:hal-00719506 From: Marie-Annick Guillemer <> Submitted on: Friday, 20 July 2012 09:19:33 Updated on: Friday, 20 July 2012 11:33:35