本文關(guān)鍵詞： 循環(huán)碼 Z_(2~k)-線性碼 Z_2Z_4碼 Z_2Z_(2~k) -加性碼 Z_2 R碼 Gray映射　出處：《安慶師范大學(xué)》2016年碩士論文　論文類型：學(xué)位論文

【摘要】：早在1948年,Claude Shannon發(fā)表了關(guān)于通信的數(shù)學(xué)理論的文章,并指出了糾錯碼的存在,此后,糾錯碼得到了迅速的發(fā)展。1957年,E.Prange首先引入了線性碼、循環(huán)碼的概念,并將此研究推廣到環(huán)上,其對剩余類環(huán)Z_k上的研究產(chǎn)生了重要的影響。自1990年以來,環(huán)Z_4和Z_(2~k)上的碼,Z_2-Z_4-加性碼,Z_2_(2~k) Z碼也越來越受關(guān)注,并且Gray映射下的二元碼的研究也取得了較大的突破。這些環(huán)上的碼已經(jīng)顯示了它在實際應(yīng)用中的良好前景。本文正是在此基礎(chǔ)上,研究了R=F_2+uF_2碼的性質(zhì),并對Z_2 R在Gray映射下的二元碼的進行了探討,并用實例說明,這種加性群在Gray映射下也能得到好的二元線性碼。本文將分為四個部分對環(huán)碼在Gray映射下的二元最優(yōu)碼進行探索。(1)緒論:概括了編碼理論的內(nèi)容,引入了糾錯碼。列出了常用的糾錯碼(如線性碼、循環(huán)碼),并給出了它們的概念和性質(zhì);(2)研究關(guān)于環(huán)Z_(2~k)上的碼,包括Z_4碼的結(jié)構(gòu)和Z_(2~k)循環(huán)碼的結(jié)構(gòu),詳細闡述了Z_4碼在Gray映射下的二元碼,并通過計算得出了成為好碼的最優(yōu)參數(shù);(3)關(guān)于Z_2_(2~k) Z碼的研究,包括Z_2_Z_4-循環(huán)碼的結(jié)構(gòu)、Z_2_(2~k) Z碼的結(jié)構(gòu)以及Z_2_Z_4碼在Gray映射下的二元碼,并通過計算得出了成為好碼的最優(yōu)參數(shù);(4)在前三部分研究的基礎(chǔ)上,我們將環(huán)碼推廣到Z_2R(R=F_2+uF_2)碼上,分別研究環(huán)R=F_2+uF_2結(jié)構(gòu)和Z_2 R碼的性質(zhì),并對Z_2 R在Gray映射下的二元碼進行進一步的研究,最終得到成為好碼的最優(yōu)參數(shù),完成環(huán)碼Gray映射下的二元最優(yōu)碼探索。
[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.
【學(xué)位授予單位】：安慶師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2016
【分類號】：O157.4

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