Strong Grِbner bases and cyclic codes over a finite-chain ring.

We present a unified theory of Gröbner bases over a principal ideal ring with a view to applications in Coding Theory, for example to the structure of codes over a finite-chain ring, e.g. a Galois ring. Our theory is effective and yields an extension of Buchbergerʹs algorithm to principal ideal rings. In particular, given a set of generators for a cyclic code over a Galois ring, we can compute a ‘minimal strong’ Gröbner basis for the code. We give some applications to cyclic codes.