On minimality of convolutional ring encoders

• Authors: Margreta Kuijper;Raquel Pinto
• Venue: IEEE Transactions on Information Theory
• Year: 2009

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 = Zpr 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 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 pγ, where γ is the sum of the row degrees of the minimal p-encoder. In particular, we show that any convolutional code over Zpr 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 Zpr admits a noncatastrophic p-encoder.