On minimality of convolutional ring encoders

TL;DR

This paper extends the concept of minimal polynomial encoders from finite fields to finite rings Z_{p^r} for convolutional codes, introducing p-encoders that produce minimal trellis representations with nonlinear features.

Contribution

It introduces the notion of minimal p-encoders for convolutional codes over Z_{p^r} and shows how to manipulate polynomial encodings to achieve minimal trellis representations.

Findings

Existence of minimal p-encoders for codes over Z_{p^r}

Delay-freeness is an encoder property, not a code property

Conjecture on noncatastrophic p-encoders for all codes over Z_{p^r}

Abstract

Convolutional codes are considered with code sequences modelled as semi-infinite Laurent series. It is wellknown that a convolutional code C over a finite group G has a minimal trellis representation that can be derived from code sequences. It is also wellknown that, for the case that G is a finite field, any polynomial encoder of C can be algebraically manipulated to yield a minimal polynomial encoder whose controller canonical realization is a minimal trellis. In this paper we seek to extend this result to the finite ring case G = Z_{p^r} by introducing a socalled "p-encoder". We show how to manipulate a polynomial encoding of a noncatastrophic convolutional code over Z_{p^r} to produce a particular type of p-encoder ("minimal p-encoder") whose controller canonical realization is a minimal trellis with nonlinear features. The minimum number of trellis states is then expressed as p^gamma, where gamma is the sum of the row degrees of the minimal p-encoder. In particular, we show that any convolutional code over Z_{p^r} admits a delay-free p-encoder which implies the novel result that delay-freeness is not a property of the code but of the encoder, just as in the field case. We conjecture that a similar result holds with respect to catastrophicity, i.e., any catastrophic convolutional code over Z_{p^r} admits a noncatastrophic p-encoder.