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

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Arithmetic of Finite Fields (WAIFI 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9061))

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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.

This research is supported by NNSF of China (61202068), Talents youth Fund of Anhui Province Universities (2012SQRL020ZD). Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (05015133) and the Project of Graduate Academic Innovation of Anhui University (N0. yfc100005).

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Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Anhui University, Hefei, China

    Yan Liu & Minjia Shi

  2. CNRS/LTCI Telecom Paris Tech, 46 rue Barrault, 75013, Paris, France

    Patrick Solé

Authors
  1. Yan Liu
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  2. Minjia Shi
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  3. Patrick Solé
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Corresponding author

Correspondence to Minjia Shi .

Editor information

Editors and Affiliations

  1. Department of Computer Science, Univ of California, Santa Barbara, Santa Barbara, California, USA

    Çetin Kaya Koç

  2. University of Paris VIII, Paris, France

    Sihem Mesnager

  3. Sabancı University, Istanbul, Turkey

    Erkay SavaĹź

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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. https://doi.org/10.1007/978-3-319-16277-5_12

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  • DOI: https://doi.org/10.1007/978-3-319-16277-5_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-16276-8

  • Online ISBN: 978-3-319-16277-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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