有限域上常循环码的研究
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摘要
设Fq是含有q=p”个元素的有限域,这里p是Fq的特征.本文主要研究Fq上常循环码的分类,几类常循环码、极小循环码的代数结构与几何结构.
全文分五章.第一章引言,我们介绍研究背景及相关工作,列出我们得到的主要结果,并给出本文所需的预备知识.
第二章：我们引进了常循环码间保距同构的概念,使得在同一个保距同构类中的常循环码有相同的代数结构及重量分布,这样就只需按保距同构类来研究常循环码.对Fq中的任意非零元λ和μ,我们给出了λ-常循环码保距同构于μ-常循环码的几个充要条件.特别地,令μ=1,我们也得到了λ-常循环码与循环码保距同构的充要条件.
第三章：根据Fq的任意非零元λ所在的保距同构类,我们给出了域Fq上长为ltps的所有λ-常循环码的生成多项式,这里l是一个与域Fq的特征p互素的素数,t,s均为非负整数.
第四章：假定s是一个使得X2s+1在Fq2[X]中能完全分解成一次因式的正整数,且L是一个与域Fq的特征p互素的奇素数.我们给出Fq上长为2slt的所有自对偶与自正交的负循环码的生成多项式.
第五章：我们研究域Fq上长为lt的极小循环码,这里l为q-1的一个素因子,t是一个正整数.我们得到了域Fq上长为lt的所有极小循环码的生成幂等元,极小Hamming距离,维数及检验多项式.
This dissertation is devoted to the classification of constacyclic codes over finite fields, and some classes constacyclic codes for their algebraic struc-tures and distance structures.
This dissertation consists of five parts. In chapter1, we introduce the background and main ideas of the present research, and then we recall the necessary notations and known results.
In chapter2, we introduce a concept "isometry" for the nonzero ele-ments of Fq to classify constacyclic codes over Fq such that the constacyclic codes belonging to the same isometry class have the same distance structures and the same algebraic structures. Some necessary and sufficient condition-s for any two elements of Fq*isometric to each other are established; as a consequence, the constacyclic codes isometric to cyclic codes are described.
In chapter3, we classify the constacyclic codes of length (?)tps over FPm. into isometry classes, characterize explicitly the polynomial generators of the constacyclic codes of each isometry class, where (?) is a prime different from the characteristic of Fq, and s, t are positive integers.
In chapter4, assuming that s is a positive integer such that X2s+1fac-tors completely into degree-one factors in Fq2[X], we obtain the polynomial generators of all self-dual and self-orthogonal negacyclic codes of length2s(?)t over Fq, where (?) is an odd prime coprime to the characteristic of Fq and t is a positive integer.
In chapter5, we study minimal cyclic codes of length (?)m over a finite field Fq, where (?) is a prime divisor of q—1and m is a positive integer. Explicit expressions for the primitive idempotents, check polynomials, mini-mum Hamming distances and the dimensions of these codes are obtained.
全文分五章.第一章引言,我们介绍研究背景及相关工作,列出我们得到的主要结果,并给出本文所需的预备知识.
第二章：我们引进了常循环码间保距同构的概念,使得在同一个保距同构类中的常循环码有相同的代数结构及重量分布,这样就只需按保距同构类来研究常循环码.对Fq中的任意非零元λ和μ,我们给出了λ-常循环码保距同构于μ-常循环码的几个充要条件.特别地,令μ=1,我们也得到了λ-常循环码与循环码保距同构的充要条件.
第三章：根据Fq的任意非零元λ所在的保距同构类,我们给出了域Fq上长为ltps的所有λ-常循环码的生成多项式,这里l是一个与域Fq的特征p互素的素数,t,s均为非负整数.
第四章：假定s是一个使得X2s+1在Fq2[X]中能完全分解成一次因式的正整数,且L是一个与域Fq的特征p互素的奇素数.我们给出Fq上长为2slt的所有自对偶与自正交的负循环码的生成多项式.
第五章：我们研究域Fq上长为lt的极小循环码,这里l为q-1的一个素因子,t是一个正整数.我们得到了域Fq上长为lt的所有极小循环码的生成幂等元,极小Hamming距离,维数及检验多项式.
This dissertation is devoted to the classification of constacyclic codes over finite fields, and some classes constacyclic codes for their algebraic struc-tures and distance structures.
This dissertation consists of five parts. In chapter1, we introduce the background and main ideas of the present research, and then we recall the necessary notations and known results.
In chapter2, we introduce a concept "isometry" for the nonzero ele-ments of Fq to classify constacyclic codes over Fq such that the constacyclic codes belonging to the same isometry class have the same distance structures and the same algebraic structures. Some necessary and sufficient condition-s for any two elements of Fq*isometric to each other are established; as a consequence, the constacyclic codes isometric to cyclic codes are described.
In chapter3, we classify the constacyclic codes of length (?)tps over FPm. into isometry classes, characterize explicitly the polynomial generators of the constacyclic codes of each isometry class, where (?) is a prime different from the characteristic of Fq, and s, t are positive integers.
In chapter4, assuming that s is a positive integer such that X2s+1fac-tors completely into degree-one factors in Fq2[X], we obtain the polynomial generators of all self-dual and self-orthogonal negacyclic codes of length2s(?)t over Fq, where (?) is an odd prime coprime to the characteristic of Fq and t is a positive integer.
In chapter5, we study minimal cyclic codes of length (?)m over a finite field Fq, where (?) is a prime divisor of q—1and m is a positive integer. Explicit expressions for the primitive idempotents, check polynomials, mini-mum Hamming distances and the dimensions of these codes are obtained.
引文
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[13]G. K. Bakshi. M. Raka, Self-dual and self-orthogonal negacyclic codes of length 2pn over a finite field, Finite Fields Appl.,19(2013),39-54.
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[23]R. T. Bilous. An enumeration of binary self-dual codes of length 32. Des, Codes Cryptogr.,2002,26,61-86.
[24]T. Blackford, Negacyclic codes over Z4 of even length, IEEE Trans. Inform. Theory,49(6)(2003),1417-1424.
[25]T. Blackford, Negacyclic duadic codes, Finite Fields Appl.14(2008),930-943.
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[30]B. Chen, L. Lin, H. Liu, Matrix Product Codes with Rosenbloom-Tsfasman Metric, Acta Math Sci., accepted.
[31]B. Chen, Y. Fan, L. Lin, H. Liu, Constacyclic codes over finite fields, Finite Fields Appl.,18(2012),1217-1231.
[32]B. Chen, H. Liu, G. Zhang, A class of minimal cyclic codes over finite fields, submitted to Designs, Codes. Cryptogr., accepted.
[33]B. Chen, H. Liu, G. Zhang, Minimal cyclic codes of length tpn, preprinted.
[34]B. Chen, H. Q. Dinh, H. Liu, repeated-root constacyclic codes of length (?)tps and their duals, preprinted.
[35]B. Chen, H. Liu, Self-dual and self-orthogonal negacyclic codes of length 2mpn over a finite field, preprinted.
[36]陈鲁生,沈世镒,编码理论基础,北京：高等教育出版社,2005.
[37]J. H. Conway, N. J. A. Sloane, Self-dual codes over the integers modulo 4, J. Comb. Theory ser.A 62(1993),30-45.
[38]P. Delsarte and J. M. Goethals, Irreducible binary cyclic codes of even dimen-sion, Chapel Hill, NC,1970,100-113.
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