【作者】 刘芳；

【导师】 游宏；

【作者基本信息】 哈尔滨工业大学 ， 基础数学， 2011， 硕士

【摘要】 循环码是线性分组码的一类重要子码,在理论和应用中都有着重要的科研价值。循环码比一般线性码拥有更多代数结构,因而引起编码和密码理论研究者的兴趣与关注。循环码的自身特性,又使得其在信息传递中更容易实现编译。随着有限域上循环码的编码理论的日益成熟,人们开始研究有限环上的循环码,已经有不少文献对剩余类环Z₄和四个元素的环F₂+u²F₂上的循环码进行了研究,但对于八元环F₂+u²F₂+u²F₂上的循环码的研究却很少。本文主要研究八元环F₂+u²F₂+uF₂上的循环码的结构和其周期分布。循环码的周期分布,是一个较新的概念,其实质就是计数问题,与循环码的质量分布、码长、信息率一样,都是循环码的参数问题。杨义先、胡正名教授于1992年首次提出纠错码的周期分布概念后,引起了编码与密码领域的学者们的关注,许多学者都对有限域Fq上的循环码的周期分布做了进一步的研究,给出了一些计算码的周期分布的公式。探究循环码的周期分布,可以给出更好的非线性循环码,构造纠错能力更强的重码和置换码等,具有实际应用价值。本文的主要工作:（1）研究了R+F₂+u²F₂+u²F₂这类因式分解不唯一环上的一元多项式分解的一些性质,证明了xⁿ+1在R[x]中关于基本多项式的分解在不计较相伴元的前提下与它在F₂[x]中的分解相同,为R+F₂+u²F₂+u²F₂上循环码的研究奠定了基础。（2）研究了R+F₂+uF₂+u²F₂上奇数长循环码的结构,给出码长为n（n为奇数）的R+循环码的个数,即为xⁿ+1分解式中基本不可约因子的个数。（3）讨论了环R+F₂+u²F₂+u²F₂上循环码的周期分布,给出奇数长循环码的周期分布计算公式。更多还原

【Abstract】 Cyclic codes are an important sub-category of linear codes, invaluable in both theory and practice. Owing to their more rigorous algebraic structure over normal linear codes, cyclic codes are given particular attention by coding theorists and cryptographers. Cyclic codes also permit easier encoding and decoding in data transfer due to their properties.As theories for cyclic codes over finite fields mature, research has begun on cyclic codes over finite rings. Current literature mostly cover cyclic codes over the residual class ring Z 4and the four-element ring F₂ + uF₂, but there exists scarce literature on the eight-element ring F₂ + u²F₂ + u²F₂. This paper mainly discusses the structure and period distribution of cyclic codes over the eight-element ring F₂ + uF₂ + u F₂.The period distribution of cyclic codes is a novel idea in coding theory. At its core is a counting problem, which is a parameter of the code much like its quality distribution, code length and information rate. After the concept of period distribution was first proposed by Professors Yixian Yang and Zhengming Hu in 1992, many coding theorists and cryptographers have conducted research on, and yielded formulas for, period distribution of cyclic codes over finite fields Fq . By investigating the period distributions of codes, one can obtain better non-linear cyclic codes, as well as weight codes and permutation codes with stronger error-correction capabilities. There exist important practical uses of period distributions.The main contributions of this paper are as follows.（1） we discuss the factorization properties of polynomials with one variable over the non-unique factorization rings R + F₂ + u²F₂ + u²F₂. We prove that x n+ 1 yields the same basic polynomial factorization in R[ x ] and F₂ [ x ], disregarding associated elements, This forms the basis of research on cyclic codes over F₂ + u²F₂ + u F₂.（2） We discuss the structure of odd-length cyclic codes over F₂ + u²F₂ + u²F₂, and give the number of R-cyclic codes of code length n . This is the number of irreducible elements in the factorization of xⁿ+ 1.（3） We discuss the period distribution of cyclic codes over F₂ + u²F₂ + u²F₂, and give the formula for calculating periodic distributions of odd-length cyclic codes.更多还原