常循环码对偶性质研究
发布时间：2019-06-03 19:09
【摘要】：循环码是一类非常重要的码.常循环码是循环码的自然推广,它保留了循环码的几乎所有良好性质.对偶性质是编码理论的重要研究对象,它在码的重量结构研究和代数结构研究等方面都有重要作用.本文主要从三个不同角度来研究常循环码(包括循环码)与对偶相关的性质.1.推广欧几里得内积和厄米特内积,对有限域的任意自同构,我们引入了伽罗瓦内积.我们用统一的方法研究最一般情形的常循环码(包括循环码,包括重根情形)的伽罗瓦自对偶性质.这个统一的方法包括：定义q-陪集函数用于刻画和构造常循环码,它推广了半单情形的用零点集刻画常循环码的方法；定义新的保距同构,它既适用于半单情形也适用于非半单情形；等等.我们得到一系列关于伽罗瓦对偶性与自对偶性的结果.特别地,对有限域的任意自同构,给出了伽罗瓦自对偶常循环码以及伽罗瓦自对偶循环码存在的充要条件.这些结果包括了常循环码的欧几里得对偶性,厄米特对偶性的有关结果作为特例.2.推广duadic循环码和Ⅱ-型duadic负循环码的概念,我们引进了even-like(也即,Ⅱ-型)和odd-like duadic常循环码的概念,并研究它们的一系列性质和存在条件.我们证明了even-like duadic常循环码是保距自正交的,并且even-like duadic常循环码的对偶码是odd-like duadic常循环码.另外,对常循环码,我们证明了当长度为n的Ⅰ-型duadic对不存在但Ⅱ-型duadic对存在时,Ⅱ-型duadic对是最大保距自正交的.随后我们给出了存在even-like duadic常循环码的充要条件,并构造了一类称作交错MDS-码的even-like duadic常循环码的例子.3.我们引进了一类新的保距同构来研究循环码的自对偶性.现有文献中的保距同构都是用模n剩余类Zn上的乘法置换来构造的,称为乘子.这类保距同构无法用于循环码保距自对偶性的研究.我们在模n剩余类Zn上用加法置换定义了一类新的保距自同构,称作平移算子.我们用这种新的方法来研究保距自对偶循环码存在的充要条件,及相关的性质.特别地,我们给出了保距自对偶循环码存在的几个等价条件.另外,我们把这种方法推广到常循环码上,并与用乘子构造Ⅰ-型duadic常循环码的方法一起来研究了保距自对偶常循环码的存在条件及相关性质.
[Abstract]:Cyclic codes are a kind of very important codes. Constant cyclic code is a natural generalization of cyclic code, which preserves almost all good properties of cyclic code. Dual property is an important research object of coding theory, which plays an important role in the study of weight structure and algebra structure of codes. In this paper, we mainly study the dual-related properties of constant cyclic codes (including cyclic codes) from three different angles. In this paper, we generalize Euclidean inner product and Hermitian inner product. For any automorphism of finite fields, we introduce Galava inner product. In this paper, we study the Galava self-duality property of the most general constant cyclic codes (including cyclic codes, including double root cases) by using a unified method. This unified method includes defining Q-coset functions to characterize and construct constant cyclic codes, which generalize the method of describing constant cyclic codes with zero sets in semi-simple cases. Define a new distance-preserving isomorphism, which is applicable to both semi-simple and non-semi-simple cases, and so on. We get a series of results on Galova duality and self-duality. In particular, for any automorphism of finite fields, the necessary and sufficient conditions for the existence of Galava self-dual constant cyclic codes and Galava self-dual cyclic codes are given. These results include the Euclidean duality of constant cyclic codes and the results of Hermitian duality as special cases. 2. In this paper, we generalize the concepts of duadic cyclic codes and type II duadic negative cyclic codes. We introduce the concepts of even-like (that is, type II) and odd-like duadic constant cyclic codes, and study a series of properties and existence conditions of them. We prove that even-like duadic constant cyclic codes are distance-preserving self-orthogonal, and the dual codes of even-like duadic constant cyclic codes are odd-like duadic constant cyclic codes. In addition, for constant cyclic codes, we prove that when the I-type duadic pair of length n does not exist but the 鈪，
本文编号：2492153
[Abstract]:Cyclic codes are a kind of very important codes. Constant cyclic code is a natural generalization of cyclic code, which preserves almost all good properties of cyclic code. Dual property is an important research object of coding theory, which plays an important role in the study of weight structure and algebra structure of codes. In this paper, we mainly study the dual-related properties of constant cyclic codes (including cyclic codes) from three different angles. In this paper, we generalize Euclidean inner product and Hermitian inner product. For any automorphism of finite fields, we introduce Galava inner product. In this paper, we study the Galava self-duality property of the most general constant cyclic codes (including cyclic codes, including double root cases) by using a unified method. This unified method includes defining Q-coset functions to characterize and construct constant cyclic codes, which generalize the method of describing constant cyclic codes with zero sets in semi-simple cases. Define a new distance-preserving isomorphism, which is applicable to both semi-simple and non-semi-simple cases, and so on. We get a series of results on Galova duality and self-duality. In particular, for any automorphism of finite fields, the necessary and sufficient conditions for the existence of Galava self-dual constant cyclic codes and Galava self-dual cyclic codes are given. These results include the Euclidean duality of constant cyclic codes and the results of Hermitian duality as special cases. 2. In this paper, we generalize the concepts of duadic cyclic codes and type II duadic negative cyclic codes. We introduce the concepts of even-like (that is, type II) and odd-like duadic constant cyclic codes, and study a series of properties and existence conditions of them. We prove that even-like duadic constant cyclic codes are distance-preserving self-orthogonal, and the dual codes of even-like duadic constant cyclic codes are odd-like duadic constant cyclic codes. In addition, for constant cyclic codes, we prove that when the I-type duadic pair of length n does not exist but the 鈪，
本文编号：2492153
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