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Self-dual skew codes and factorization of skew polynomials

Delphine Boucher 1, Félix Ulmer 1

(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

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From: Marie-Annick Guillemer <>

Submitted on: Friday, 20 July 2012 09:19:33

Updated on: Friday, 20 July 2012 11:33:35