【作者】 张妍；

【导师】 董学东；

【作者基本信息】 辽宁师范大学 ， 应用数学， 2004， 硕士

【摘要】 Hammons等人证明了一些十分重要的二元非线性码是Z4上的线性码在Gray映射下的像,这之后针对有限环Z4和Zpm(p为一个素数,m ≥ 1)上的码的研究逐步开展起来,并获得了很多重要结果. 本论文主要目的是将关于Zpm上码的一些结果推广到有限Artin局部主理想环上.首先,我们回顾了Artin环的定义及其基本性质, 然后讨论了有限Artin局部主理想环的一些性质.其次,我们深入研究有限Artin局部主理想环上的循环码. 先给出Hensel提升定理及循环码Hensel提升的定义;然后详细讨论有限Artin局部主理想环上的循环码的结构,推导出它们的生成元的形状,指出在满足一定条件下这样的码可以由单个生成元生成. 由此证明了在一定条件下,Rn = R[x]/ xn ? 1 是一个主理想环,这里R是一个有限Artin局部主理想环. 最后初步讨论了有限Artin局部主理想环上循环码的幂等元问题.更多还原

【Abstract】 Hammons et.al have proved that some important binary nonlinearcodes are images under the Gray map of linear codes over Z4. Since then peopleall around the world are interested in studying codes over ?nite rings Z4 andZpm,where p is a odd prime and m ≥ 1. Many important results on codes overZ4 and Zpm have been obtained. The main purpose of this thesis is to generalizesome results on codes over Z4 and Zpm to a ?nite Artin local principal idealring (PIR). Firstly, we review some de?nitions and several basic results of Artin ring;then we discuss some properties of ?nite Artin local PIR. Next, we focus on our attention to cyclic codes over ?nite Artin localPIR. We ?rst give the Hensel lifting and the de?nition of Hensel lift of a cycliccode. Then we give more detailed discuss on the structure of cyclic codes over?nite Artin local PIR. We show that such codes may be produced by a singlegenerator. In particular, we proves that the ring Rn = R[x]/ xn?1 is principal,where R is a ?nite Artin local PIR. Finally the idempotent generators of cycliccodes over ?nite Artin local PIR are considered.更多还原