(Sigma-Delta) Codes

TL;DR

This paper introduces cyclic ($f(t),\sigma,\delta$)-codes in Ore polynomial rings, generalizing previous $ heta$-codes, and provides their construction, properties, and matrix representations, including special cases related to Wedderburn polynomials.

Contribution

It generalizes $ heta$-codes to cyclic ($f(t),\sigma,\delta$)-codes in Ore rings and characterizes their control matrices, especially for Wedderburn polynomials.

Findings

Control matrices are given by generalized Vandermonde matrices.

All Wedderburn polynomials in $ ext{F}_q[t; heta]$ are characterized.

Pseudo-linear transformations are key to the code structure.

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.