基于经典Goppa码的非对称量子稳定子码构造

Constructions of asymmetric quantum stabilizer codes based on classical Goppa codes

自从Calderbank等人建立了从经典纠错码构造量子纠错码的CRSS构造法以来,人们利用经典纠错码构造了大量的性能良好的量子纠错码,称为量子稳定子码.最近的物理实验表明,大多数量子力学系统中发生量子比特翻转错误的概率远小于量子相位翻转错误的概率,针对这一情况所构造的纠错码称为非对称量子纠错码.本文分别基于嵌套包含Goppa码与对偶包含Goppa码构造了一系列新的非对称量子稳定子码.在基于嵌套包含Goppa码构造非对称量子码时,首先对Goppa码的选取做一定的限制,以便解析构造量子码.对于一般情况下的构造,则是借助于数学软件Matlab计算Goppa码对偶码的最小距离进行的.在基于对偶包含Goppa码的构造中,所构造量子码的纠错能力主要体现在纠正Z类型错误上.

Since Calderbank et al. established CRSS construction, which constructs quantum error-correcting codes （QECC） from classical error-correcting codes, many QECC demonstrating good performance have been constructed h-om classical error-correcting codes; these codes are referred to as quantum stabilizer codes. Recent physical experiments indicate that the probabilities of occurrence of qubit-fiip errors are much less than those of phase-flip errors in most quantum mechanical systems. Error-correcting codes thus constructed are referred to as asymmetric quantum codes. In this paper, series of asymmetric quantum stabilizer codes are constructed using nested Goppa codes and dual contained Goppa codes. In the construction of a.symmetric quantum codes based on nested Goppa codes, the choice of Goppa codes is limited to construct codes with analysis methods. In nornml circmnstances, the construction is executed with the help of mathematical software Matlab, which is used to compute the mininmm distance of Goppa codes. In the construction of asynlmetric quantum codes based on dual contained Goppa codes, the error correction ability of the codes constructed in this paper is reflected in the correction of Z-type errors.