環(huán)上具有互補對偶的碼
發(fā)布時間：2018-10-08 17:26
【摘要】：編碼理論是數(shù)學(xué)、信息論和工程的交叉學(xué)科,它在通信(例如衛(wèi)星的信號傳輸、數(shù)據(jù)存儲等)中有著廣泛的應(yīng)用。為了使通信系統(tǒng)具有更好的檢錯和糾錯能力,通常需要對發(fā)出信息進行編碼。編碼理論的研究起源于1948年Shannon的《通信的數(shù)學(xué)理論》一文。編碼理論最早的研究只限于有限域上,人們首先研究了二元域上的碼,逐漸將其推廣到任意有限域上的碼。隨著有限域上的編碼理論研究成果的豐富,人們開始研究環(huán)上的編碼理論,1994年,A.R.Hammons等學(xué)者證明了域上某些二元非線性碼可以視為四元線性碼在Gray映射下的像,自此四元環(huán)和更一般的有限環(huán)上的編碼理論成為編碼理論研究領(lǐng)域的一個熱點。具有互補對偶的碼(簡稱LCD碼)是一類非常重要的線性碼。1992年Massey率先研究了有限域上的LCD碼,證明了生成矩陣為G的碼是LCD碼當(dāng)且僅當(dāng)矩陣GGT是可逆的。但環(huán)上LCD碼的結(jié)構(gòu)非常復(fù)雜,目前為止人們僅對一些特殊環(huán)(例如鏈環(huán))上的LCD碼進行了研究,本文試圖對其它環(huán)上的LCD碼的判別條件進行研究。本文主要研究兩類環(huán)(鏈環(huán)Z4、非鏈環(huán)Fq+vFq)上的LCD碼。研究方法主要通過Gray映射將環(huán)上的碼轉(zhuǎn)化為域上的碼,借助碼的生成矩陣的標(biāo)準(zhǔn)形,給出有限鏈環(huán)Z4和有限非鏈環(huán)Fq+Fq上碼是LCD碼的一些充分條件和必要條件,并在適當(dāng)?shù)臈l件下給出碼是LCD碼的充要條件。
[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.
【學(xué)位授予單位】：北京交通大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O157.4;O153.3
本文編號：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.
【學(xué)位授予單位】：北京交通大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O157.4;O153.3
【參考文獻】
相關(guān)期刊論文 前4條
1 毛琪莉;劉修生;;基于環(huán)F_4+uF_4上的線性碼[J];數(shù)學(xué)雜志;2013年02期
2 梁華;唐元生;;有限鏈環(huán)上循環(huán)碼的研究[J];計算機工程;2010年24期
3 李�？�,楊繼明,張寶華;關(guān)于剩余類環(huán)Z_m上的線性方程組[J];陜西師范大學(xué)學(xué)報(自然科學(xué)版);2004年S1期
4 段煉;剩余類環(huán)Z_m上的自由模Z_m~((n))[J];河南大學(xué)學(xué)報(自然科學(xué)版);2001年02期
，本文編號：2257644
本文鏈接：http://sikaile.net/shoufeilunwen/benkebiyelunwen/2257644.html
最近更新
教材專著
