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

  • Amit SharmaEmail author
  • Maheshanand Bhaintwal
Original Paper
  • 101 Downloads

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.

Keywords

Skew polynomial rings Skew-cyclic codes 2D skew-cyclic codes Automorphisms Generating sets 

Notes

Acknowledgements

This work was partially supported by DST, Govt. of India, under Grant No. SB/S4/MS: 893/14. Also, the first author would like to thank the Council of Scientific & Industrial Research (CSIR), India for providing financial support.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Indian Institute of Technology RoorkeeRoorkeeIndia

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