【作者】 刘艳；

【导师】 刘春雷；

【作者基本信息】 上海交通大学 ， 基础数学， 2015， 博士

【摘要】 循环码是一类特殊的线性分组码.循环码构造简单且具有很好的代数结构从而便于分析.除此之外,循环码的编码和译码都可以利用移位寄存器来实现.而且,循环码具有高效的编码和译码算法.因此,循环码在通信和存储系统中都有广泛的应用.循环码的重量分布可以给出这个码的最小距离,从而可以给出这个码的纠错能力.不仅如此,利用某些解码算法来检错和纠错时,通过循环码的重量分布还可以估计发生错误的概率.因此,确定循环码的重量分布在理论和实践方面都有很重要的意义.前人在研究循环码的重量分布方面已经得到了很多重要的结论.在他们的思想启发下,本文构造了三类可约循环码,并确定了这三类循环码的重量分布.本文的具体内容可概括如下第一章简要介绍了本文的研究背景以及循环码重量分布的研究现状,同时介绍了本文的主要研究内容和相关的预备知识.第二章构造了一类Frt上的可约循环码C1,其校验多项式为π、(一π)-1和π(pk+1)/2在Fpt上的最小多项式的最小公倍式.通过计算得出,C1是一个参数为[pm-1,3m0,pt-1/2pm-t]的6-重循环码.不仅如此,事实上,我们确定了该类循环码的重量分布.这里,p是一个奇素数,π是有限域Frmm的一个本原元.其中,m是一个正的奇数,k是一个正整数,使得s=m/d≥3.这里,d=gcd(m,k),t是整除d的任意一个正整数,m0=m/t.第三章构造了一类Fpt上的可约循环码C2,其校验多项式为π、π(pk+1)和π(p2k+1)在Fpt上的最小多项式的最小公倍式.经计算得出,该码是参数为[pm-1,3m0, (pt-1)(pm-t-pm+3d-2t/2)]的5-重循环码.事实上,本文在第三章完全确定了该类循环码的重量分布.这里的p和π如上所述.其中,m和k均为正整数使得s=m/d≥5是一个奇数.这里,d=gcd(m,k).t是整除d的一个正整数使得d/t是一个奇数,m0=m/t.第四章构造了一类Fpt上的可约循环码C3,其校验多项式为π-1、π-2、π-(pk+1)和π(pp2k+1)在Fpt上的最小多项式的最小公倍式,并得出该码是Fpt上的参数为[pm-1,4m0,(pt-1)pm-t-pm+4d-t]的循环码.该类循环码的重量分布在本文第四章被完全确定.这里对m、k、d、t、m0、p和π的限制如第三章.更多还原

【Abstract】 Cyclic codes are a subclass of linear block codes. They can be easily designed and have good algebraic structure that are convenient for analysis. Besides, they can be effi-ciently encoded using shift registers technically. Moreover, they have efficient encoding and decoding algorithms. So cyclic codes have wide applications in communication and storage systems. The weight distribution of a cyclic code gives the minimum distance of it and, thus, the error correcting capability. In addition, the weight distribution can be used to estimate the error probability of error detection and correction with respect to some decoding algorithms. Therefore, determining the weight distributions of cyclic codes is not only a problem of theoretical interest, but also of practical importance. There are many remarkable results on the weight distributions of cyclic codes. Inspired by these original ideas, three classes of reducible cyclic codes are constructed and their weight distributions are determined in this paper. The main content of this paper can be generalized as the following.In Chapter one, we introduce the research backgrounds, some results on the weight distributions of cyclic codes, and then list the main work and some preliminaries.In Chapter two, let m be an odd positive integer and k be any positive integer such that s=m/d≥3, where d=gcd(m, k). Let t be a positive divisor of d and m0=m/t. Under these conditions, a class of reducible cyclic codes C1 over Fpt is constructed, whose parity-check polynomial is the least common multiple of the minimal polynomials of π-1, (-π)-1 and π-(p+1)/2 over Fpt, where p is an odd prime and π is a primitive element of m. By calculation, C1 is a cyclic code over Fpt with parameters [pm 1,3m0,2-pt-1/2pm-t]. Besides, the weight distribution of C1 is determined.In Chapter three, let m and k be any two positive integers such that s=m/d≥5 is odd, where d=gcd(m,k). Let t be a positive divisor of d such that d/t is odd, m0=m/t. Let p and π be defined as above. Then a class of reducible cyclic codes C2 over Fpt is constructed, whose parity-check polynomial is the least common multiple of the minimal polynomials of π-2,π-(pk+1) and π-(p2k+1) over Fpt Moreover, C2 is proved to be a cyclic code over Fpt with parameters [pm-1,3m0,(pt-1)(pm-t-pm+3d-2t)]. Furthermore, the weight distribution of C2 is determined.In Chapter four, let m, k, d, t, m0, p and π be defined as in Chapter three. A class of reducible cyclic codes C3 over Fpt is constructed, whose parity-check polynomial is the least common multiple of the minimal polynomials of π-1, π-2,π-(pk+1) and π-(p2k+1) over Fpt. Moreover, in this chapter, C3 is proved to be a cyclic code over Fpt with parameters [pm-1,4m0,(pt-1)pm-t-pm+4d-t/2]. Furthermore,the weight distribution of C3 is determined.更多还原