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Two families of BCH codes and new quantum codes

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Abstract

In this paper, two families of non-narrow-sense (NNS) BCH codes of lengths \(n=\frac{q^{2m}-1}{q^2-1}\) and \(n=\frac{q^{2m}-1}{q+1}\) (\(m\ge 3)\) over the finite field \(\mathbf {F}_{q^2}\) are studied. The maximum designed distances \(\delta ^\mathrm{new}_\mathrm{max}\) of these dual-containing BCH codes are determined by a careful analysis of properties of the cyclotomic cosets. NNS BCH codes which achieve these maximum designed distances are presented, and a sequence of nested NNS BCH codes that contain these BCH codes with maximum designed distances are constructed and their parameters are computed. Consequently, new nonbinary quantum BCH codes are derived from these NNS BCH codes. The new quantum codes presented here include many classes of good quantum codes, which have parameters better than those constructed from narrow-sense BCH codes, negacyclic and constacyclic BCH codes in the literature.

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Acknowledgements

The authors are very grateful to the reviewers and the Editor-in-Chief, Prof. Weinstein, for their detailed comments and suggestions that much improved the presentation and quality of this paper.

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Authors and Affiliations

  1. Department of Basic Sciences, Air Force Engineering University, Xi’an, 710051, Shaanxi, People’s Republic of China

    Hao Song, Ruihu Li, Junli Wang & Yang Liu

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  1. Hao Song
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  2. Ruihu Li
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  3. Junli Wang
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  4. Yang Liu
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Corresponding author

Correspondence to Ruihu Li.

Additional information

This work is supported by the National Natural Science Foundation of China under Grant No. 11471011 and Natural Science Foundation of Shaanxi province under Grant No. 2017JQ1032.

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Song, H., Li, R., Wang, J. et al. Two families of BCH codes and new quantum codes. Quantum Inf Process 17, 270 (2018). https://doi.org/10.1007/s11128-018-2042-3

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  • DOI: https://doi.org/10.1007/s11128-018-2042-3

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