arXiv:1304.6518v1 [math.RA] 24 Apr 2013 (σ, δ )-codes M’Hammed Boulagouaz and Andr´ e Leroy ∗ University of King Khalid, Abha, Saudi Arabia E-mail: boulag@yahoo.com Universit´ e d’Artois, Facult´ e Jean Perrin Rue Jean Souvraz 62 307 Lens, France E-mail: leroy@euler.univ-artois.fr Abstract: In this paper we introduce the notion of cyclic (f (t), σ, δ )-codes for f (t) ∈ A[t; σ, δ ]. These codes generalize the θ-codes as introduced by D. Boucher, F. Ulmer, W. Geiselmann [2]. We construct generic and control matrices for these codes. As a particular case the (σ, δ )-W -code associated to a Wedderburn polynomial are defined and we show that their control matrices are given by generalized Van- dermonde matrices. All the Wedderburn polynomials of F q [t; θ] are described and their control matrices are presented. A key role will be played by the pseudo-linear transformations. 1 Introduction and preliminaries The use of rings in coding theory started when it appears that working over rings allowed certain codes to be looked upon as linear codes. The use of noncommuative rings emerged recently in coding theory due to the pertinence of Frobenius rings for generalizing Mac Williams theorems (cf. [17], for details) and also because of the use of Ore polynomial rings as source of generalizations of cyclic codes (cf. e.g. [2],[4],[16]). With some few exceptions (e.g. [13],[3]) the Ore polynomial rings used so far in coding theory are mainly of automorphisms type with a (finite) field as base ring. This paper shows how one can use general Ore extensions to not only define codes, but as well give their generic and control matrices. Factorizations techniques in Ore polynomial rings play an important role in these questions and the interested reader can consult [6], [9] or [10] for more information on this matter. Since they are intimately related to modules over Ore extensions and to factorizations, the pseudo- * 1