The dual-containing primitive BCH codes with the maximum designed distance and their applications to quantum codes

  • Xueying ShiEmail author
  • Qin Yue
  • Yansheng Wu


Classical Bose–Chaudhuri–Hocquenghem (BCH) codes are an important class of cyclic codes in terms of their error-correcting capability. BCH codes that contain their Euclidean (or Hermitian) dual codes have been widely used in constructing quantum stabilizer codes. In this paper, necessary and sufficient conditions on Euclidean and Hermitian dual-containing primitive BCH codes with the maximum designed distance are shown, and necessary conditions on non-narrow-sense primitive BCH codes that do not contain their Euclidean or Hermitian dual codes are given. The results of Euclidean and Hermitian dual-containing primitive BCH codes in (Aly et al. IEEE Trans Inf Theory 53(3):1183–1188, 2007) are extended. Moreover, the dimensions of some non-narrow-sense primitive BCH codes are determined and some quantum codes are constructed.


Cyclic code BCH code Dual-containing 

Mathematics Subject Classification

94B15 81P70 



This research is supported by National Natural Science Foundation of China (No. 61772015), Postgraduate Research & Practice Innovation Program of Jiangsu Province (Nos. KYCX17_0225 and KYCX18_0241), and the Fundamental Research Funds for the Central Universities (No. NZ2018005). The paper is also supported by Foundation of Science and Technology on Information Assurance Laboratory (No. KJ-17-010), the Natural Science Foundation of Jiangsu Higher Education Institutions of China (Grant No. 17KJB110018), and Guangxi Natural Science Foundation(2016GXNSFDA380017). Additionally, the authors are grateful to the Editor and the anonymous referees for their useful comments and suggestions which helped to improve the presentation of this paper.


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

Authors and Affiliations

  1. 1.Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjingPeople’s Republic of China
  2. 2.State Key Laboratory of CryptologyBeijingPeople’s Republic of China

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