On Minimality of Convolutional Ring Encoders

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 = Zpr by introducing a socalled "p-encoder". We show how to manipulate a polynomial encoding 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 p γ , 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.

MDS Convolutional Codes Over a Finite Ring

“…We use the techniques presented in the previous sections. These techniques were used in [4] to introduce some definitions such as -encoder, -indices, and -degree which are equivalent to those in field case.…”

“…Recall that . Definition 4.1 [4]: Let be a convolutional code of length over . Let be a polynomial matrix whose rows are -linearly independent -generator sequence.…”

“…We also give an explicit construction as well as examples of block codes that meet the Singleton bound. The main result in this paper is presented in Section IV, where we use the concepts of -encoder, -dimension, -degree, and -indices for a convolutional code over finite rings, introduced in [4]. The main objective is to provide a Singleton bound for convolutional codes over finite rings which generalizes the one in the field case.…”

Feedback equivalence of convolutional codes over finite rings

“…However, convolutional codes over rings do not behave in the same way as convolutional codes over elds because their behavior depends strongly on the structure of the underlying ring. The study of properties, encoders, p-basis and dual convolutional codes over nite rings, has been developed in [13][14][15][16] among others. In [17], a bound on the free distance of convolutional codes over Z p r was developed generalizing the results described in [7].…”

Gröbner bases and behaviors over finite rings

“…Recent papers [13], [16], [14], [15] consider behaviors over the ring Z p r , where p is a prime integer and r is a positive integer. In these papers the theory of [17] is put to work to extend the above two minimal realization problems to systems over the ring Z p r .…”