(Sigma-Delta) Codes

Abstract

In this paper we introduce the notion of cyclic ($f(t),\sigma ,\delta$)-codes for $f(t)\in \Ore$. These codes generalize the $\theta$-codes as introduced by D. Boucher, F. Ulmer, W. Geiselmann \cite{BGU}. We construct generic and control matrices for these codes. As a particular case the ($\si,\de$)-$W$-code associated to a Wedderburn polynomial are defined and we show that their control matrices are given by generalized Vandermonde matrices. All the Wedderburn polynomials of ${\mathbb{F}}_q[t;\theta ]$ are described and their control matrices are presented. A key role will be played by the pseudo-linear transformations.

15 Pages Submitted version October 2012