自正交码的构造及其应用

发布时间：2018-05-01 05:13

本文选题：循环码 + Gray映射　；参考：《合肥工业大学》2017年硕士论文

【摘要】：有限域与有限环上自正交码一直是纠错码理论研究的重要课题。随着量子纠错技术的不断发展,人们发现可以利用经典的自正交码来构造量子码,从而引起学者对构造经典自正交码产生浓厚兴趣。本文,研究了环F_q+uF_q上的循环自正交码的结构(其中u~2=0)。基于F_q+uF_q上上的循环自正交码,参数最优的量子码被构造出来。同时,我们又利用F_(4~m)上的厄米特自正交常循环码构造出了量子最大距离可分(MDS)码。文章包含以下三个部分:1.对环F_2+uF_2上的循环自正交码进行了研究。得到F_2+uF_2上循环自正交码的生成多项式,且计算出F_2+uF_2上奇长度的循环自正交码。2.对域F_(4~m)上任意长度的厄米特自正交常循环码的结构进行了研究。通过F_(4~m)上厄米特自正交常循环码的生成多项式,得出F_(4~m)上厄米特自正交常循环码的存在条件,确立了F_(4~m)上厄米特自正交常循环码的计数公式。并且利用F_(4~m)上偶长度的厄米特自正交常循环码构造出了量子最大距离可分(MDS)码。3.给出了F_q+uF_q上上的循环自正交码存在的一个充分必要条件(其中q(?)1(mod 4))。通过构造一个从F_q+uF_q上到F_q~2的Gray映射,使得可以由F_q+uF_q上上的循环自正交码得到F_q上的自正交码。通过这种方法参数最优的量子码被构造出来。
[Abstract]:Self-orthogonal codes over finite fields and finite rings have always been an important subject in the theory of error-correcting codes. With the development of quantum error correction technology, it is found that classical self-orthogonal codes can be used to construct quantum codes. In this paper, the structure of cyclic self-orthogonal codes over FQ uF_q is studied. Based on cyclic self-orthogonal codes over FQ uF_q, quantum codes with optimal parameters are constructed. At the same time, we construct quantum maximum distance divisible MDS codes by using Hermitian self-orthogonal cyclic codes on Flives. The article consists of the following three parts: 1. Cyclic self orthogonal codes over F 2 uF_2 are studied. The generating polynomials of cyclic self-orthogonal codes on F _ s _ 2 uF_2 are obtained, and the cyclic self-orthogonal codes. 2 of odd length on F _ s _ 2 uF_2 are calculated. The structure of Hermitian self-orthogonal constant cyclic codes of arbitrary length over the field FStup 4 m is studied. By means of the generating polynomial of Hermitian self-orthogonal regular cyclic codes on FSP 4m, the existence conditions of Hermitian self-orthogonal regular cyclic codes on FSP 4m) are obtained, and the counting formula of Hermitian self-orthogonal constant cyclic codes on FSP 4m) is established. The quantum maximum distance divisible MDS) code. 3 is constructed by using the Hermitian self-orthogonal constant cyclic code with even length on the F _ S _ 4 / m. A necessary and sufficient condition for the existence of cyclic self-orthogonal codes on FQ uF_q is given. By constructing a Gray map from Fq uF_q to F_q~2, the self-orthogonal codes on FQ uF_q can be obtained by cyclic self-orthogonal codes on FQ uF_q. Quantum codes with optimal parameters are constructed by this method.
【学位授予单位】：合肥工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157.4

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