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28-05-2019 | Original Paper | Issue 6/2019

Applicable Algebra in Engineering, Communication and Computing 6/2019

A class of 2D skew-cyclic codes over \({\mathbb {F}}_{q}+u{\mathbb {F}}_{q}\)

Journal:
Applicable Algebra in Engineering, Communication and Computing > Issue 6/2019
Authors:
Amit Sharma, Maheshanand Bhaintwal
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Abstract

In this paper we present a class of 2D skew-cyclic codes over \(R={\mathbb {F}}_{q}+u{\mathbb {F}}_{q}, u^2=1\), using the bivariate skew polynomial ring \(R[x,y,\theta ,\sigma ]\), where \({\mathbb {F}}_q\) is a finite field, and \(\theta \) and \(\sigma \) are two commuting automorphisms of R. After defining a partial order on \(R[x,y,\theta ,\sigma ],\) we obtain division algorithm for \(R[x,y,\theta ,\sigma ]\) under two different conditions. The structure of 2D skew-cyclic codes over R is obtained in terms of their generating sets. For this, we have classified these codes into different classes, based on certain conditions they satisfy, and accordingly obtained their generating sets in each case separately. A decomposition of a 2D skew-cyclic code C over R into 2D skew-cyclic codes over \({\mathbb {F}}_{q}\) is studied and some examples are given to illustrate the results.

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