环上具有互补对偶的码
发布时间：2018-10-08 17:26
【摘要】：编码理论是数学、信息论和工程的交叉学科,它在通信(例如卫星的信号传输、数据存储等)中有着广泛的应用。为了使通信系统具有更好的检错和纠错能力,通常需要对发出信息进行编码。编码理论的研究起源于1948年Shannon的《通信的数学理论》一文。编码理论最早的研究只限于有限域上,人们首先研究了二元域上的码,逐渐将其推广到任意有限域上的码。随着有限域上的编码理论研究成果的丰富,人们开始研究环上的编码理论,1994年,A.R.Hammons等学者证明了域上某些二元非线性码可以视为四元线性码在Gray映射下的像,自此四元环和更一般的有限环上的编码理论成为编码理论研究领域的一个热点。具有互补对偶的码(简称LCD码)是一类非常重要的线性码。1992年Massey率先研究了有限域上的LCD码,证明了生成矩阵为G的码是LCD码当且仅当矩阵GGT是可逆的。但环上LCD码的结构非常复杂,目前为止人们仅对一些特殊环(例如链环)上的LCD码进行了研究,本文试图对其它环上的LCD码的判别条件进行研究。本文主要研究两类环(链环Z4、非链环Fq+vFq)上的LCD码。研究方法主要通过Gray映射将环上的码转化为域上的码,借助码的生成矩阵的标准形,给出有限链环Z4和有限非链环Fq+Fq上码是LCD码的一些充分条件和必要条件,并在适当的条件下给出码是LCD码的充要条件。
[Abstract]:Coding theory is an interdisciplinary subject of mathematics, information theory and engineering. It is widely used in communication (such as satellite signal transmission, data storage, etc.). In order to make the communication system have better error-detection and error-correcting capability, it is usually necessary to encode the outgoing information. The research of coding theory originates from the article "Mathematical Theory of Communication" by Shannon in 1948. The earliest study of coding theory is limited to finite fields. Firstly, codes over binary fields are studied, which are gradually extended to codes on arbitrary finite fields. With the enrichment of the research results of coding theory on finite fields, people began to study the coding theory over rings. In 1994, some bivariate nonlinear codes were proved to be the images of quaternion linear codes under Gray maps, such as A. R. Hammons, and some other scholars proved that some binary nonlinear codes on the domain can be regarded as images of quaternion linear codes under Gray mapping. Since then, coding theory over quaternion rings and more general finite rings has become a hot topic in the field of coding theory. Codes with complementary duality (LCD codes for short) are a class of very important linear codes. In 1992, Massey first studied LCD codes over finite fields, and proved that the codes with generated matrix G are LCD codes if and only if the matrix GGT is reversible. However, the structure of LCD codes over rings is very complex. So far, only the LCD codes on some special rings (such as chain rings) have been studied. This paper attempts to study the criteria of LCD codes on other rings. In this paper, we study the LCD codes of two kinds of rings (chain ring Z _ 4, non-linked ring Fq vFq). In this paper, by means of Gray mapping, the codes on the ring are transformed into codes on the domain. With the help of the canonical form of the generating matrix of the codes, some sufficient and necessary conditions for the codes over the finite chain ring Z4 and the finite non-chained ring Fq to be LCD codes are given. A sufficient and necessary condition for the code to be LCD code is given under appropriate conditions.
【学位授予单位】：北京交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157.4;O153.3
本文编号：2257644
[Abstract]:Coding theory is an interdisciplinary subject of mathematics, information theory and engineering. It is widely used in communication (such as satellite signal transmission, data storage, etc.). In order to make the communication system have better error-detection and error-correcting capability, it is usually necessary to encode the outgoing information. The research of coding theory originates from the article "Mathematical Theory of Communication" by Shannon in 1948. The earliest study of coding theory is limited to finite fields. Firstly, codes over binary fields are studied, which are gradually extended to codes on arbitrary finite fields. With the enrichment of the research results of coding theory on finite fields, people began to study the coding theory over rings. In 1994, some bivariate nonlinear codes were proved to be the images of quaternion linear codes under Gray maps, such as A. R. Hammons, and some other scholars proved that some binary nonlinear codes on the domain can be regarded as images of quaternion linear codes under Gray mapping. Since then, coding theory over quaternion rings and more general finite rings has become a hot topic in the field of coding theory. Codes with complementary duality (LCD codes for short) are a class of very important linear codes. In 1992, Massey first studied LCD codes over finite fields, and proved that the codes with generated matrix G are LCD codes if and only if the matrix GGT is reversible. However, the structure of LCD codes over rings is very complex. So far, only the LCD codes on some special rings (such as chain rings) have been studied. This paper attempts to study the criteria of LCD codes on other rings. In this paper, we study the LCD codes of two kinds of rings (chain ring Z _ 4, non-linked ring Fq vFq). In this paper, by means of Gray mapping, the codes on the ring are transformed into codes on the domain. With the help of the canonical form of the generating matrix of the codes, some sufficient and necessary conditions for the codes over the finite chain ring Z4 and the finite non-chained ring Fq to be LCD codes are given. A sufficient and necessary condition for the code to be LCD code is given under appropriate conditions.
【学位授予单位】：北京交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157.4;O153.3
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