【作者】 张萍；

【导师】 李锦；

【作者基本信息】 合肥工业大学 ， 应用数学， 2021， 硕士

【摘要】 BCH码是一类重要的循环码,它有高效的编码和译码算法,因而有着广泛的应用。常循环BCH码是BCH码的发展和推广,它有着更灵活的参数和良好的代数结构,可以用于构造参数优良的量子纠错码。本文主要研究了有限域Fq2上一类码长为n=(q2m-1)/[r(q-1)]的λ-常循环BCH码,并利用该类常循环码构造了 一些参数较好的量子码,其中ord(λ)=r,r|(q+1),q是素数幂。具体研究内容如下:一、根据m的奇偶性分类研究了q2模nr的分圆陪集的结构,给出了该类常循环BCH码是厄米特对偶包含码的最大设计距离,确定了该类厄米特对偶包含常循环BCH码的维数,利用厄米特构造得到了在相同长度和设计距离下,比量子BCH码维数更大的量子常循环BCH码。二、当m≥3且m为奇数时,通过分解该类常循环BCH码的定义集,计算出了纠缠态,利用纠缠辅助量子码的构造方法,构造出了一类纠缠辅助量子码。更多还原

【Abstract】 BCH codes are an important class of cyclic codes with efficient encoding and decoding algorithms,so they have widely applications.Constacyclic BCH codes are the development and generalization of BCH codes,they have more flexible parameters and good algebraic structures.Constacyclic BCH codes can be used to construct quantum error correcting codes with good parameters.In this paper,we mainly study a class ofλ-constacyclic BCH codes over the finite field Fq2 with length n=q2m-1/[r(q-1)],and we use these constacyclic BCH codes to construct some quantum codes with good parameters,where ord(λ)=r,r|(q+1)and q is a prime power.Specific research contents are as follows:一、The structures of q2 cyclotomic cosets modulo nr are studied according to the parity of m,the maximum design distance of this class of constacyclic BCH codes to be Hermitian dual-containing codes are given,the dimensions of these Hermitian dual-containing constacyclic BCH codes are determined,and some quantum constacyclic BCH codes are obtained by using Hermitian construction,they have larger dimensions than the quantum BCH codes at the same lengths and design distances.二、For m≥3 and m is odd,the number of entangled state is calcuated by decomposing the defining set of this class of constacyclic BCH codes,and a class of entanglement-assisted quantum codes is constructed by using the construction method of entanglement-assisted quantum codes.更多还原