State spaces of convolutional codes, codings and encoders

TL;DR

This paper presents a concise theoretical framework for convolutional codes and encoders, establishing a connection between their abstract spaces and providing proofs for encoder size bounds and minimality conditions.

Contribution

It introduces a unified theory of abstract spaces for convolutional codes and encoders, clarifying their relationship and simplifying key proofs.

Findings

The size of a convolutional encoder is at least its inner degree.

Minimal encoders have size equal to the external degree when the matrix is reduced.

The paper clarifies the relationship between code and encoder spaces.

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.