On Minimality of Convolutional Ring Encoders

Abstract

Convolutional codes are considered with code sequences modeled as semi-infinite Laurent series. It is well known 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 well known 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 = ℤp r by introducing a so-called "p-encoder". We show how to manipulate a polynomial encoding scheme of a noncatastrophic convolutional code over ℤpr to produce ..