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A class of narrow-sense BCH codes over \(\mathbb {F}_q\) of length \(\frac{q^m-1}{2}\)

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Abstract

BCH codes with efficient encoding and decoding algorithms have many applications in communications, cryptography and combinatorial design. This paper studies a class of linear codes of length \( \frac{q^m-1}{2}\) over \(\mathbb {F}_q\) with special trace representation, where q is an odd prime power. With the help of the inner distributions of some subsets of association schemes of quadratic forms, we determine the weight enumerators of these codes. Determining some cyclotomic coset leaders \(\delta _i\) of cyclotomic cosets modulo \( \frac{q^m-1}{2}\), we prove that narrow-sense BCH codes of length \( \frac{q^m-1}{2}\) with designed distance \(\delta _i=\frac{q^m-q^{m-1}}{2}-1-\frac{q^{ \lfloor \frac{m-3}{2} \rfloor +i}-1}{2}\) have the corresponding trace representation, and have the minimal distance \(d=\delta _i\) and the Bose distance \(d_B=\delta _i\), where \(1\le i\le \lfloor \frac{m+11}{6} \rfloor \).

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Acknowledgements

This work was supported by SECODE project and the National Natural Science Foundation of China (Grant Nos. 11871058, 11531002, 11701129). C. Tang also acknowledges support from 14E013, CXTD2014-4 and the Meritocracy Research Funds of China West Normal University. Y. Qi also acknowledges support from Zhejiang provincial Natural Science Foundation of China (LQ17A010008, LQ16A010005).

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Authors and Affiliations

  1. School of Mathematics and Information, China West Normal University, Nanchong, 637002, China

    Xin Ling & Chunming Tang

  2. Department of Mathematics, University of Paris VIII, 93526, Saint-Denis, France

    Sihem Mesnager

  3. LAGA UMR 7539, CNRS, Sorbonne Paris Cité, University of Paris XIII, 93430, Paris, France

    Sihem Mesnager

  4. Telecom ParisTech, 75013, Paris, France

    Sihem Mesnager

  5. School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang, 310018, China

    Yanfeng Qi

  6. Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

    Chunming Tang

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  1. Xin Ling
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  2. Sihem Mesnager
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  3. Yanfeng Qi
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  4. Chunming Tang
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Correspondence to Chunming Tang.

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Communicated by Q. Xiang.

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Ling, X., Mesnager, S., Qi, Y. et al. A class of narrow-sense BCH codes over \(\mathbb {F}_q\) of length \(\frac{q^m-1}{2}\). Des. Codes Cryptogr. 88, 413–427 (2020). https://doi.org/10.1007/s10623-019-00691-0

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  • DOI: https://doi.org/10.1007/s10623-019-00691-0

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