State spaces of convolutional codes, codings and encoders

Abstract

In this paper we give a compact presentation of the theory of abstract spaces for convolutional codes and convolutional encoders, and show a connection between them that seems to be missing in the literature. We use it for a short proof of two facts: the size of a convolutional encoder of a polynomial matrix is at least its inner degree, and the minimal encoder has the size of the external degree if the matrix is reduced.

Submitted to IEEE Transactions on Information Theory