  In this paper, we mainly consider quasi-cyclic (QC) codes over finite chain
rings. We study module structures and trace representations of QC codes, which
lead to some lower bounds on the minimum Hamming distance of QC codes.
Moreover, we investigate the structural properties of 1-generator QC codes.
Under some conditions, we discuss the enumeration of 1-generator QC codes and
describe how to obtain the one and only one generator for each 1-generator QC
code.
