Construction of new quantum codes via Hermitian dual-containing matrix-product codes

Abstract

In 2001, Blackmore and Norton introduced an important tool called matrix-product codes, which turn out to be very useful to construct new quantum codes of large lengths. To obtain new and good quantum codes, we first give a general approach to construct matrix-product codes being Hermitian dual-containing and then provide the constructions of such codes in the case \(s{\mid }(q^{2}-1)\), where s is the number of the constituent codes in a matrix-product code. For \(s{\mid } (q+1)\), we construct such codes with lengths more flexible than the known ones in the literature. For \(s{\mid } (q^{2}-1)\) and \(s{\not \mid } (q+1)\), such codes are constructed in an unusual manner; some of the constituent codes therein are not required to be Hermitian dual-containing. Accordingly, by Hermitian construction, we present two procedures for acquiring quantum codes. Finally, we list some good quantum codes, many of which improve those available in the literature or add new parameters.

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Acknowledgements

The authors would like to express their sincere gratefulness to the anonymous referees for their valuable comments and suggestions which have highly improved the quality of this manuscript. This work was supported by National Natural Science Foundation of China under Grant No. 11271217.

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Correspondence to Jianlian Cui.

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Project supported by the National Natural Science Foundation of China (Grant No. 11271217).

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Cao, M., Cui, J. Construction of new quantum codes via Hermitian dual-containing matrix-product codes. Quantum Inf Process 19, 427 (2020). https://doi.org/10.1007/s11128-020-02921-0

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Keywords

  • Hermitian dual-containing codes
  • Matrix-product codes
  • Generalized Reed–Solomon codes
  • Extended generalized Reed–Solomon codes
  • Quantum codes