基于环码Gray映射下的二元最优码探索

发布时间：2018-01-27 16:01

本文关键词： 循环码 Z_(2~k)-线性码 Z_2Z_4码 Z_2Z_(2~k) -加性码 Z_2 R码 Gray映射　出处：《安庆师范大学》2016年硕士论文　论文类型：学位论文

【摘要】：早在1948年,Claude Shannon发表了关于通信的数学理论的文章,并指出了纠错码的存在,此后,纠错码得到了迅速的发展。1957年,E.Prange首先引入了线性码、循环码的概念,并将此研究推广到环上,其对剩余类环Z_k上的研究产生了重要的影响。自1990年以来,环Z_4和Z_(2~k)上的码,Z_2-Z_4-加性码,Z_2_(2~k) Z码也越来越受关注,并且Gray映射下的二元码的研究也取得了较大的突破。这些环上的码已经显示了它在实际应用中的良好前景。本文正是在此基础上,研究了R=F_2+uF_2码的性质,并对Z_2 R在Gray映射下的二元码的进行了探讨,并用实例说明,这种加性群在Gray映射下也能得到好的二元线性码。本文将分为四个部分对环码在Gray映射下的二元最优码进行探索。(1)绪论:概括了编码理论的内容,引入了纠错码。列出了常用的纠错码(如线性码、循环码),并给出了它们的概念和性质;(2)研究关于环Z_(2~k)上的码,包括Z_4码的结构和Z_(2~k)循环码的结构,详细阐述了Z_4码在Gray映射下的二元码,并通过计算得出了成为好码的最优参数;(3)关于Z_2_(2~k) Z码的研究,包括Z_2_Z_4-循环码的结构、Z_2_(2~k) Z码的结构以及Z_2_Z_4码在Gray映射下的二元码,并通过计算得出了成为好码的最优参数;(4)在前三部分研究的基础上,我们将环码推广到Z_2R(R=F_2+uF_2)码上,分别研究环R=F_2+uF_2结构和Z_2 R码的性质,并对Z_2 R在Gray映射下的二元码进行进一步的研究,最终得到成为好码的最优参数,完成环码Gray映射下的二元最优码探索。
[Abstract]:As early as 1948, Claude Shannon published an article on the mathematical theory of communication, and pointed out the existence of error-correcting codes. Since then, error-correcting codes have developed rapidly. 1957. E. Prange first introduced the concept of linear code and cyclic code, and extended this research to ring, which has an important influence on the research of residual class ring ZK. Since 1990. The number Zs / ZS _ 2-ZZ _ 4 _ _ _. And the research of binary codes under Gray mapping has also made a great breakthrough. The codes on these rings have shown a good prospect in practical applications. This paper is based on this. In this paper, we study the properties of the uF_2 code RF2s, and discuss the binary code of Zs _ 2R under the Gray mapping, and illustrate it with an example. This additive group can also obtain good binary linear codes under Gray mappings. In this paper, we will divide into four parts to explore the binary optimal codes of ring codes under Gray mappings. Introduction: the content of coding theory is summarized. The error-correcting codes are introduced. The commonly used error-correcting codes (such as linear codes, cyclic codes) are listed, and their concepts and properties are given. In this paper, we study the code on Zs _ s _ 2k), including the structure of Zs _ 4 code and the structure of Z _ s _ T _ 2k) cyclic code, and elaborate the binary code of Z _ s _ 4 code under Gray mapping in detail. The optimal parameters of good code are obtained by calculation. (3) Research on the Z2S _ 2s _ 2K) Z code, including the structure of the Zs _ 2s _ s _ _ _. The structure of Z code and the binary code of Zs _ 2S _ 4 code under Gray mapping, and the optimum parameters for making a good code are obtained. 4) on the basis of the first three parts, we extend the ring code to the Z2RZR / RX / F2uF2) code, and study the properties of the ring RGF _ 2 uF_2 structure and Z2R _ 2R code, respectively. Furthermore, the binary code of Zs _ 2R under Gray mapping is further studied. Finally, the optimal parameter of good code is obtained, and the exploration of binary optimal code under Gray mapping of ring code is completed.
【学位授予单位】：安庆师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O157.4

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