Quadratic Residue Codes over \(\mathbb {F}_p+v\mathbb {F}_p+v^{2}\mathbb {F}_p\)

Conference paper

DOI: 10.1007/978-3-319-16277-5_12

Part of the Lecture Notes in Computer Science book series (LNCS, volume 9061)
Cite this paper as:
Liu Y., Shi M., Solé P. (2015) Quadratic Residue Codes over \(\mathbb {F}_p+v\mathbb {F}_p+v^{2}\mathbb {F}_p\). In: Koç Ç., Mesnager S., Savaş E. (eds) Arithmetic of Finite Fields. WAIFI 2014. Lecture Notes in Computer Science, vol 9061. Springer, Cham

Abstract

This article studies quadratic residue codes of prime length \(q\) over the ring \(R=\mathbb {F}_{p}+v\mathbb {F}_{p}+v^{2}\mathbb {F}_{p},\) where \(p,q\) are distinct odd primes. After studying the structure of cyclic codes of length \(n\) over \(R,\) quadratic residue codes over \(R\) are defined by their generating idempotents and their extension codes are discussed. Examples of codes and idempotents for small values of \(p\) and \(q\) are given. As a by-product almost MDS codes over \(\mathbb {F}_7\) and \(\mathbb {F}_{13}\) are constructed.

Keywords

Cyclic codes Quadratic residue codes Generating idempotents Dual codes 

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.School of Mathematical SciencesAnhui UniversityHefeiChina
  2. 2.CNRS/LTCI Telecom Paris TechParisFrance

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