Linear codes over finite rings and modules

The foundations of linear code theory over finite rings and modules are developed. The main objects of investigation are: systematic code, dual code, McWilliams identity, parity-check matrix an the Hamming distance of a code. The properties of codes over modules and linear spaces are compared, and the representations of linear codes by polylinear recurrences are described, the latter being the most efficient for systematic and Abelian group codes. The special role of quasi-Frobenius modules in code theory is revealed. As corollaries we obtain and generalize some known results. In particular, we build cyclic Hamming and BCH codes over an arbitrary primary module.