State Space Realizations and Monomial Equivalence for Convolutional Codes

TL;DR

This paper investigates the relationship between state space realizations and code equivalence in convolutional codes, establishing conditions under which codes are monomially equivalent based on their adjacency matrices.

Contribution

It proves that minimal state space realizations are equivalent under full state feedback if they define the same code and characterizes monomial equivalence via adjacency matrices for codes with positive Forney indices.

Findings

Minimal realizations belong to the same code iff they are full state feedback equivalent.

Codes with positive Forney indices are monomially equivalent iff they share the same adjacency matrix.

Adjacency matrices encode complete weight information of the codewords.

Abstract

We will study convolutional codes with the help of state space realizations. It will be shown that two such minimal realizations belong to the same code if and only if they are equivalent under the full state feedback group. This result will be used in order to prove that two codes with positive Forney indices are monomially equivalent if and only if they share the same adjacency matrix. The adjacency matrix counts in a detailed way the weights of all possible outputs and thus contains full information about the weights of the codewords in the given code.