Cyclic Codes Over Finite Artin Local Principal Ideal Rings

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.