Xing Lijuan, Li Zhuo
The State Key Laboratory of Integrated Services Networks, Xidian University, Xi'an 710071, China .
Entropy (Basel). 2021 Jun 3;23(6):712. doi: 10.3390/e23060712.
Quantum error correcting codes (QECCs) play an important role in preventing quantum information decoherence. Good quantum stabilizer codes were constructed by classical error correcting codes. In this paper, Bose-Chaudhuri-Hocquenghem (BCH) codes over finite fields are used to construct quantum codes. First, we try to find such classical BCH codes, which contain their dual codes, by studying the suitable cyclotomic cosets. Then, we construct nonbinary quantum BCH codes with given parameter sets. Finally, a new family of quantum BCH codes can be realized by Steane's enlargement of nonbinary Calderbank-Shor-Steane (CSS) construction and Hermitian construction. We have proven that the cyclotomic cosets are good tools to study quantum BCH codes. The defining sets contain the highest numbers of consecutive integers. Compared with the results in the references, the new quantum BCH codes have better code parameters without restrictions and better lower bounds on minimum distances. What is more, the new quantum codes can be constructed over any finite fields, which enlarges the range of quantum BCH codes.
量子纠错码（QECCs）在防止量子信息退相干方面起着重要作用。良好的量子稳定器码是由经典纠错码构造而成的。在本文中，有限域上的博斯 - 乔杜里 - 霍克文黑姆（BCH）码被用于构造量子码。首先，我们通过研究合适的分圆陪集来寻找包含其对偶码的经典BCH码。然后，我们构造具有给定参数集的非二进制量子BCH码。最后，通过斯蒂恩对非二进制卡尔德班克 - 肖尔 - 斯蒂恩（CSS）构造和厄米特构造的扩展，可以实现一个新的量子BCH码族。我们已经证明分圆陪集是研究量子BCH码的良好工具。定义集包含最多数量的连续整数。与参考文献中的结果相比，新的量子BCH码具有更好的码参数且无限制，并且在最小距离上有更好的下界。此外，新的量子码可以在任何有限域上构造，这扩大了量子BCH码的范围。
