Minimal Trellis Construction for Finite Support Convolutional Ring Codes

Abstract

We address the concept of “minimal polynomial encoder” for finite support linear convolutional codes over ${\mathbb Z}_{p^r}$. These codes can be interpreted as polynomial modules which enables us to apply results from the 2007-paper [8] to introduce the notions of “p-encoder” and “minimal p-encoder”. Here the latter notion is the ring analogon of a row reduced polynomial encoder from the field case. We show how to construct a minimal trellis representation of a delay-free finite support convolutional code from a minimal p-encoder. We express its number of trellis states in terms of a degree invariant of the code. The latter expression generalizes the wellknown expression in terms of the degree of a delay-free finite support convolutional code over a field to the ring case. The results are also applicable to block trellis realization of polynomial block codes over ${\mathbb Z}_{p^r}$, such as CRC codes over ${\mathbb Z}_{p^r}$.