有限链环上线性码深度分布的研究

发布时间：2018-10-30 11:37

【摘要】：随着环上线性码的结构及性质的深入研究,研究者们将对有限域上线性码深度的研究推广到了对环上线性码深度的研究,本文主要从线性码的生成多项式以及生成矩阵的角度,研究了环上线性码的深度谱及深度分布,具体如下：首先,给出了环R=Fpk+uFpk上码字深度的概念,利用该环上循环码的结构再结合有限域上循环码的生成多项式,确定了线性码r0(C)与r1(C)的结构及性质,从而得到了环R上任意长度循环码的深度谱和深度分布,最后研究了自对偶循环码存在的条件；其次,在循环码深度分布研究的基础上,研究了环R上任意长度的负循环码及其自对偶码的深度谱,并给出了长为pm的负循环码深度分布；再次,给出了R=Fq+uFq+u2Fq上线性码r0(C)、r1(C)和r2(C)的定义,根据有限域上线性码深度谱的计算方法,研究了R上负循环码的深度谱,并且给出了其深度分布公式,也给出了自对偶负循环码的深度分布公式；最后,研究了环R'=F2+uF2+u2F2+u3F2上线性码C的深度谱,利用R'上线性码C的标准生成矩阵,分别得到了线性码r0(C)、r1(C)、r2(C)和r3(C)的生成矩阵,证明了C的深度谱中至少含有k0+k1+k2+k3个非零数值,给出了线性码深度分布的一般公式。
[Abstract]:With the further study of the structure and properties of linear codes over a ring, researchers have extended the study of the depth of linear codes over finite fields to the depth of linear codes over rings. In this paper, the generation polynomial of linear codes and the generation matrix of linear codes are introduced. The depth spectrum and depth distribution of linear codes over a ring are studied. Firstly, the concept of codeword depth over a ring R=Fpk uFpk is given, and the structure of cyclic codes over the ring is combined with the generating polynomial of cyclic codes over finite fields. The structure and properties of linear codes r0 (C) and r 1 (C) are determined, and the depth spectrum and depth distribution of cyclic codes of arbitrary length on the ring R are obtained. Finally, the existence conditions of self dual cyclic codes are studied. Secondly, based on the research of depth distribution of cyclic codes, the depth spectrum of negative cyclic codes and their self-dual codes of arbitrary length over ring R is studied, and the depth distribution of negative cyclic codes with length of pm is given. Thirdly, the definitions of linear code r 0 (C), r 1 (C) and r 2 (C) on R=Fq uFq u2Fq are given. According to the calculation method of depth spectrum of linear code over finite field, the depth spectrum of negative cyclic code over R is studied, and the depth distribution formula is given. The depth distribution formula of self-dual negative cyclic codes is also given. Finally, the depth spectrum of linear code C over ring R'=F2 uF2 u2F2 u3F2 is studied. By using the standard generation matrix of linear code C on R', the generating matrices of linear code r 0 (C), r 1 (C), r 2 (C) and r 3 (C) are obtained, respectively. It is proved that the depth spectrum of C contains at least K 0 k 1 k 2 k 3 nonzero values, and the general formula of depth distribution of linear codes is given.
【学位授予单位】：合肥工业大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O157.4

【相似文献】