有限域上的一类量子码

A class of quantum codes over finite fields

在Avanti Ketkar等工作的基础上,进一步研究给出了有限域上的另一类类似BCH码的经典码,并证明与该经典码相对应的[[N,K,D]]q量子码和[[N+1,K-1,D+1]]q(q≥2)扩展量子码都存在.在二元域上构造扩展量子码的过程主要采用了偶校验,其运算在内积上进行;在非二元域上构造扩展量子码的过程主要采用了使得行向量各个元素相加为0的方法,并借助了有限域上本原元的性质,其运算在Hermitian内积上进行.研究结论扩展了利用经典码构建量子码的范围,证明了扩展量子码的最小距离为D+1,并给出了有关经典非二元码校验位的构造及其相关纯量子码存在的构造性证明方法.分析表明,[[N+1,K-1,D+1]]q扩展量子码比[[N,K,D]]q量子码更适宜于信息的传递.

Based on the work of Avanti Ketkar et al.,a class of classical codes similar to BCH codes over finite fields is given,and the existence of both corresponding quantum codes [[N,K,D]]q and extended quantum codes [[N＋1,K-1,D＋1]]q（q≥2） of those classical codes is proved.On the binary field,the construction process of extended quantum codes mainly uses parity check,and the operation is carried out on inner product.On the nonbinary field,the sum of every element of the vector is set to 0 in the construction process of extended quantum codes,the character of primitive element is used in the construction and the operation is carried out on Hermitian inner product.Research results expend the range of the construction of quantum codes using classical codes and prove that the minimum distance of extended quantum codes is equal to D＋1.The construction of check bits of classical nonbinary codes and the constructive proof of existence of related pure quantum codes are given.Analyses show that extended quantum codes [[N＋1,K-1,D＋1]]q are more suitable for transmitting message than quantum codes [[N,K,D]]q.