1-Generator quasi-cyclic codes over Fpm+uFpm+...+us−1Fpm

In this paper, we mainly consider the quasi-cyclic (QC) codes over finite chain ring R=Fpm+uFpm+⋯+us−1Fpm, where p is a prime number and m, s are positive integers such that s≥2 and us=0. We investigate the structural properties of the QC codes over R, regarding them as subcodes of cyclic codes over some Galois extension rings of R. This point of view leads to construct the 1-generator QC codes with index s over finite field Fpm. Further, we study the structural properties of annihilators of the 1-generator QC codes. For the case s=2, under the conditions gcd(n,p)=1 and gcd(|pm|n,l)=1, we discuss the enumeration of the distinct 1-generator QC codes and describe how to obtain one and the only one generator for each 1-generator QC code. Finally, we give some examples to illustrate the main work in this paper.