Let p be a prime and let n,k be integers, gcd(p,n) = 1 and k > 1. Calderbank and Sloane characterised cyclic codes of length n over the integers modulo p^k in a form which intuitively suggested a 'minimal strong Groebner basis (SGB)' over the integers modulo p^k.

Let D be a principal ideal domain. The structure of a minimal SGB for an ideal of D[x₁,...,x_n] is due to Becker and Weisspfenning, Lazard and others. We outline an effective theory of minimal SGB's for ideals of R[x₁,...,x_n], where R is an arbitrary principal ideal ring (e.g. the integers modulo p^k), generalising the PID case.

We then characterise cyclic codes of arbitrary length over R using minimal SGB's. This yields the result of Calderbank and Sloane as a special case.

This is joint work with Ana Salagean, Nottingham Trent University, UK.