| Codes over finite fields have already been studied deeply, and codes over finite rings have been paid more attention among coding researchers. However, Research on Quasi-Cyclic Codes and Quasi-Twisted Codes over finite chain rings was still limited. In this paper, we study the structure of Quasi-Cyclic Codes and Quasi-Twisted Codes with arbitrary lengths over the finite chain ringFq+uFq(u2=0).The decomposition of Quasi-Cyclic Codes over Fp+uFp+u2Fp (u3=0) are also discussed in this paper.1. The form of generators and the minimal generating sets of Quasi-Cyclic Codes with arbitrary lengths over Fq+uFq (u2=0) are determined, where q=ps,p is a prime number and s is a positive integer. We also study the structure of Generalized Quasi-Cyclic Codes with arbitrary lengths over Fp+uFp+u2Fp (u3=0). The decomposition and the minimal generating sets of Generalized Quasi-Cyclic Codes are given. Meanwhile we determine the lower bounds on the minimum distance of the free1-generator Generalized Quasi-Cyclic Codes. Furthmore, we use another point of view to research a special class of Generalized Quasi-Cyclic Codes called Quasi-Cyclic Codes over the ringFp+uFp+u2Fp(u3=0).2.The structure of (1+λu)-Quasi-Twisted Codes with arbitrary lengths over Fq+uFq (u2=0) are investigated, where q=ps, p is a prime number, s is a positive integer and λ is a unit over the ring. We give the generators and the lower bounds on the minimum distance of (1+λu)-Quasi-Twisted Codes with arbitrary lengths. Particularly, we discuss the structure of (1+λu)-Quasi-Twisted Codes when λ=1. We also give the minimum generating sets of three types of (1+λu)-Quasi-Twisted Codes over Fq+uFq (u2=0). |
