二元量子码的截短码和扩展码的构造

Construction of punctured and extended quantum codes over GF(2)

本文提出了一类从包含其对偶码的经典线性码[n,k,d]2到纯量子码[[n-i,2k-n+i,d-i]]2(0 i d-2)和[[n+1,2k-n-1,d+1]]2(或[[n+1,2k-n-1,d]]2)的基于矩阵初等变换的构造方法.基于初等代数理论,设计了其构造体系;基于检验矩阵与线性码最小距离之间的关系,分析了其纠错性能;基于稳定子体系,构建了其编译码网络.该方法正确可行的构造性证明简单,易懂,易于计算机及各种硬件系统实现.理论结果显示出该方法对一类量子码的构造非常实用.

Based on elementary transformation, we proposed a new method for constructing a class of pure quantum codes [[n - 1; 2k - n ＋ i; d - i]]2（0 ≤ i ≤ d - 2） and [[n ＋ 1; 2k - n - 1; d ＋ 1]]2（[In ＋ 1; 2k - n - 1; d]]2） from a class of classical linear codes In; k; d]2 which contain their dual codes. The construction process was designed based on the elementary algebra; the error-correcting performance of the quantum codes was analyzed based on the relationship between the parity-check matrix and the minimum distance of the classical linear codes; the encoding and decoding networks were constructed based on the stabilizer. This method is simple, straightforward and easy to implement via computer and other hardwares. The theoretical results showed that this method is practical for the construction of a class of quantum codes.