A class of constacyclic BCH codes and new quantum codes

Article

Abstract

Constacyclic BCH codes have been widely studied in the literature and have been used to construct quantum codes in latest years. However, for the class of quantum codes of length \(n=q^{2m}+1\) over \(F_{q^2}\) with q an odd prime power, there are only the ones of distance \(\delta \le 2q^2\) are obtained in the literature. In this paper, by a detailed analysis of properties of \(q^{2}\)-ary cyclotomic cosets, maximum designed distance \(\delta _\mathrm{{max}}\) of a class of Hermitian dual-containing constacyclic BCH codes with length \(n=q^{2m}+1\) are determined, this class of constacyclic codes has some characteristic analog to that of primitive BCH codes over \(F_{q^2}\). Then we can obtain a sequence of dual-containing constacyclic codes of designed distances \(2q^2<\delta \le \delta _\mathrm{{max}}\). Consequently, new quantum codes with distance \(d > 2q^2\) can be constructed from these dual-containing codes via Hermitian Construction. These newly obtained quantum codes have better code rate compared with those constructed from primitive BCH codes.

Keywords

Negacyclic code Constacyclic code Quantum code Cyclotomic coset 

Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China under Grant No.11471011 and Natural Science Foundation of Shaanxi under Grant No.2015JM1023.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of ScienceAir Force Engineering UniversityXi’anPeople’s Republic of China
  2. 2.The First Aeronautical College of Air ForceXinyangPeople’s Republic of China

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