三类可约循环码的重量分布

发布时间：2018-06-22 00:47

  本文选题：有限域 + 线性码　； 参考：《上海交通大学》2015年博士论文

【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.
[Abstract]:Cyclic codes are a special class of linear block codes. Cyclic codes are simple to construct and have good algebraic structure to facilitate analysis. In addition, cyclic codes can be encoded and decoded by shift registers. Moreover, cyclic codes have efficient coding and decoding algorithms. Therefore, cyclic codes are widely used in communication and storage systems. The weight distribution of the cyclic code can give the minimum distance of the code, thus the error correction ability of the code can be obtained. Moreover, when some decoding algorithms are used to detect and correct errors, the probability of errors can be estimated by the weight distribution of cyclic codes. Therefore, determining the weight distribution of cyclic codes is of great significance in both theory and practice. Many important conclusions have been obtained in studying the weight distribution of cyclic codes. In this paper, we construct three reducible cyclic codes and determine their weight distribution. The main contents of this paper can be summarized as follows: in the first chapter, the research background and the current situation of cyclic code weight distribution are briefly introduced. At the same time, the main research contents and related preparatory knowledge are introduced. In chapter 2, we construct a class of reducible cyclic codes C _ 1 on Frt, whose check polynomials are the least common times of the least polynomial of 蟺, (蟺) -1 and 蟺 (PK 1) / 2 on FPT. By calculation, it is found that C _ 1 is a 6- repeat cyclic code with a parameter of [pm-1n3m0m0m0m-1 / 2pm-t]. Moreover, in fact, we determine the weight distribution of this kind of cyclic codes. Here p is an odd prime and 蟺 is a primitive of the finite field Frmm. Where m is a positive odd number k is a positive integer, such that s=m/d 鈮，

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