On Minimality of Convolutional Ring Encoders

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 = BBZpr by introducing a so-called ldquo p-encoderrdquo. We show how to manipulate a polynomial encoding scheme of a noncatastrophic convolutional code over BBZpr to produce a particular type of p-encoder (ldquominimal p -encoderrdquo) 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 BBZpr 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 BBZpr admits a noncatastrophic p-encoder.