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

  • Xin Ling
  • Sihem Mesnager
  • Yanfeng Qi
  • Chunming TangEmail author


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 \(1\le i\le \lfloor \frac{m+11}{6} \rfloor \).


Linear code BCH code Association scheme The weight distribution Quadratic form 

Mathematics Subject Classification

94B05 94B15 05E30 15A63 



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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and InformationChina West Normal UniversityNanchongChina
  2. 2.Department of MathematicsUniversity of Paris VIIISaint-DenisFrance
  3. 3.LAGA UMR 7539, CNRS, Sorbonne Paris Cité, University of Paris XIIIParisFrance
  4. 4.Telecom ParisTechParisFrance
  5. 5.School of ScienceHangzhou Dianzi UniversityHangzhouChina
  6. 6.Department of MathematicsThe Hong Kong University of Science and TechnologyHong KongChina

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