($\sigma,\delta$)-codes

M'Hammed Boulagouaz; André Leroy

doi:10.3934/amc.2013.7.463

Article Contents

2013, Volume 7, Issue 4: 463-474. Doi: 10.3934/amc.2013.7.463

This issue Previous Article Quotients of orders in cyclic algebras and space-time codes Next Article Correlation of binary sequence families derived from the multiplicative characters of finite fields

($\sigma,\delta$)-codes

M'Hammed Boulagouaz^1, and
André Leroy^2,

1.
University of King Khalid, Abha
2.
Université d'Artois, Lille Nord de France, Rue Jean Souvraz, Lens, 62300

Received: September 30, 2012

Published: October 2013

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Abstract

In this paper we introduce the notion of cyclic ($f(t),\sigma,\delta$)-codes for $f(t)\in A[t;\sigma,\delta]$. These codes generalize the $\theta$-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 ($\sigma,\delta$)-$W$-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.

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Mathematics Subject Classification: Primary: 94B05, 94B15; Secondary: 16S36.

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