BCH码的定义集分解及应用

Decomposition of Defining Sets of BCH Codes and Its Applications

以分圆陪集理论和方法为基础,由二元码的Euclid正交性理论和四元码的Hermite正交性理论,分别引入二元BCH码和四元BCH码的定义集分解概念;再利用BCH码的定义集分解导出二元BCH码和四元BCH码的对偶码的正交分解。在此基础上,研究并解决了本原二元和四元BCH码的定义集分解;依据BCH码的定义集分解结论,构造出一些参数优良的纠缠辅助量子纠错码。定义集分解方法简化了由BCH码构造纠缠辅助量子纠错码的理论推导,改进了已有文献中确定最优纠缠比特数的算法,提供了一种计算最优纠缠比特数的新思路,为研究由循环码构造纠缠辅助量子纠错码问题提供了可借鉴的新理论和新方法。

Based on basic theory of cyclotomic coset, the concepts of decomposition of defining sets for bina ry and quaternary BCH codes are introduced respectively this decomposition of defining sets builds up a bridge among the Euclidean orthogonal decomposition of binary BCH codes and Hermitian orthogonal de composition of quaternary BCH codes. Then the orthogonal decomposition of the dual codes of binary and quaternary BCH codes is also presented. Furtherly, the decomposition of the defining sets of primitive bi- nary and quaternary BCH codes with given designed distances is well studied and solved. Applying the re- sults, some entanglement-assisted quantum codes with good parameters are constructed. The method pro- posed in this paper devised a new scheme in determining the optimal number of entangled bits, which can simplify the theoretical derivation in constructing entanglement-assisted quantum error correcting codes from classical BCH codes, and also be useful for studying general constructions of entanglement-assisted quantum error correcting codes from cyclic codes.