On Minimality of Convolutional Ring Encoders
Margreta Kuijper, Raquel Pinto · IEEE Transactions on Information Theory · 2009
Convolutional codes are considered with code sequences modeled as semi-infinite Laurent series. It is well known that a convolutional codeCover a finite groupGhas a minimal trellis representation that can be derived from code sequences. It is also well known that, for the case thatGis a finite field, any polynomial encoder ofCcan 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 caseG= \BBZprby introducing a so-called ldquop-encoderrdquo. We show how to manipulate a polynomial encoding scheme of a noncatastrophic convolutional code over\BBZprto produce a particular type ofp-encoder (ldquominimalp-encoderrdquo) whose controller canonical realization is a minimal trellis with nonlinear features. The minimum number of trellis states is then expressed aspgamma, wheregammais the sum of the row degrees of the minimalp-encoder. In particular, we show that any convolutional code over\BBZpradmits a delay-freep-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\BBZpradmits a noncatastrophicp-encoder.