Linear codes using skew polynomials with automorphisms and derivations

Delphine Boucher (1) , Félix Ulmer (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

In this work the de nition of codes as modules over skew polynomial rings of automorphism type is generalized to skew polynomial rings whose multiplication is de ned using an automorphism and an inner derivation. This produces a more gen- eral class of codes which, in some cases, produce better distance bounds than skew module codes constructed only with an automorphism. Extending the approach of Gabidulin codes, we introduce new notions of evaluation of skew polynomials with derivations and the corresponding evaluation codes. We propose several ap- proaches to generalize Reed Solomon and BCH codes to module skew codes and for two classes we show that the dual of such a Reed Solomon type skew code is an evaluation skew code. We generalize a decoding algorithm due to Gabidulin for the rank matrix and derive families of MDS and MRD codes.

Mots clés: Error-correcting codes; Decoding; Finite fields; Skew polynomial rings

Domaines: Théorie de l'information et codage [math.IT]; Théorie de l'information [cs.IT]

Delphine Boucher, Félix Ulmer. Linear codes using skew polynomials with automorphisms and derivations. Designs, Codes and Cryptography, 2014, 70 (3), pp.405-431. ⟨10.1007/s10623-012-9704-4⟩. ⟨hal-00597127⟩