本文選題：有限域 + 線性碼��； 參考：《上海交通大學》2015年博士論文

【摘要】：循環(huán)碼是一類特殊的線性分組碼.循環(huán)碼構(gòu)造簡單且具有很好的代數(shù)結(jié)構(gòu)從而便于分析.除此之外,循環(huán)碼的編碼和譯碼都可以利用移位寄存器來實現(xiàn).而且,循環(huán)碼具有高效的編碼和譯碼算法.因此,循環(huán)碼在通信和存儲系統(tǒng)中都有廣泛的應用.循環(huán)碼的重量分布可以給出這個碼的最小距離,從而可以給出這個碼的糾錯能力.不僅如此,利用某些解碼算法來檢錯和糾錯時,通過循環(huán)碼的重量分布還可以估計發(fā)生錯誤的概率.因此,確定循環(huán)碼的重量分布在理論和實踐方面都有很重要的意義.前人在研究循環(huán)碼的重量分布方面已經(jīng)得到了很多重要的結(jié)論.在他們的思想啟發(fā)下,本文構(gòu)造了三類可約循環(huán)碼,并確定了這三類循環(huán)碼的重量分布.本文的具體內(nèi)容可概括如下第一章簡要介紹了本文的研究背景以及循環(huán)碼重量分布的研究現(xiàn)狀,同時介紹了本文的主要研究內(nèi)容和相關的預備知識.第二章構(gòu)造了一類Frt上的可約循環(huán)碼C1,其校驗多項式為π、(一π)-1和π(pk+1)/2在Fpt上的最小多項式的最小公倍式.通過計算得出,C1是一個參數(shù)為[pm-1,3m0,pt-1/2pm-t]的6-重循環(huán)碼.不僅如此,事實上,我們確定了該類循環(huán)碼的重量分布.這里,p是一個奇素數(shù),π是有限域Frmm的一個本原元.其中,m是一個正的奇數(shù),k是一個正整數(shù),使得s=m/d≥3.這里,d=gcd(m,k),t是整除d的任意一個正整數(shù),m0=m/t.第三章構(gòu)造了一類Fpt上的可約循環(huán)碼C2,其校驗多項式為π、π(pk+1)和π(p2k+1)在Fpt上的最小多項式的最小公倍式.經(jīng)計算得出,該碼是參數(shù)為[pm-1,3m0, (pt-1)(pm-t-pm+3d-2t/2)]的5-重循環(huán)碼.事實上,本文在第三章完全確定了該類循環(huán)碼的重量分布.這里的p和π如上所述.其中,m和k均為正整數(shù)使得s=m/d≥5是一個奇數(shù).這里,d=gcd(m,k).t是整除d的一個正整數(shù)使得d/t是一個奇數(shù),m0=m/t.第四章構(gòu)造了一類Fpt上的可約循環(huán)碼C3,其校驗多項式為π-1、π-2、π-(pk+1)和π(pp2k+1)在Fpt上的最小多項式的最小公倍式,并得出該碼是Fpt上的參數(shù)為[pm-1,4m0,(pt-1)pm-t-pm+4d-t]的循環(huán)碼.該類循環(huán)碼的重量分布在本文第四章被完全確定.這里對m、k、d、t、m0、p和π的限制如第三章.
[Abstract]:Cyclic codes are a special class of linear block codes. Cyclic codes are simple to construct and have good algebraic structure to facilitate analysis. In addition, cyclic codes can be encoded and decoded by shift registers. Moreover, cyclic codes have efficient coding and decoding algorithms. Therefore, cyclic codes are widely used in communication and storage systems. The weight distribution of the cyclic code can give the minimum distance of the code, thus the error correction ability of the code can be obtained. Moreover, when some decoding algorithms are used to detect and correct errors, the probability of errors can be estimated by the weight distribution of cyclic codes. Therefore, determining the weight distribution of cyclic codes is of great significance in both theory and practice. Many important conclusions have been obtained in studying the weight distribution of cyclic codes. In this paper, we construct three reducible cyclic codes and determine their weight distribution. The main contents of this paper can be summarized as follows: in the first chapter, the research background and the current situation of cyclic code weight distribution are briefly introduced. At the same time, the main research contents and related preparatory knowledge are introduced. In chapter 2, we construct a class of reducible cyclic codes C _ 1 on Frt, whose check polynomials are the least common times of the least polynomial of 蟺, (蟺) -1 and 蟺 (PK 1) / 2 on FPT. By calculation, it is found that C _ 1 is a 6- repeat cyclic code with a parameter of [pm-1n3m0m0m0m-1 / 2pm-t]. Moreover, in fact, we determine the weight distribution of this kind of cyclic codes. Here p is an odd prime and 蟺 is a primitive of the finite field Frmm. Where m is a positive odd number k is a positive integer, such that s=m/d 鈮，

本文編號：2050777