On the weight distribution of convolutional codes

TL;DR

This paper investigates the weight distribution of convolutional codes using the adjacency matrix of their state diagram, proving its invariance and its role in classifying certain codes.

Contribution

It demonstrates that the adjacency matrix is an invariant of convolutional codes and uniquely determines one-dimensional binary codes up to monomial equivalence.

Findings

The adjacency matrix is an invariant of the code.

Codes with the same adjacency matrix share the same dimension and Forney indices.

For one-dimensional binary codes, the adjacency matrix uniquely determines the code.

Abstract

Detailed information about the weight distribution of a convolutional code is given by the adjacency matrix of the state diagram associated with a controller canonical form of the code. We will show that this matrix is an invariant of the code. Moreover, it will be proven that codes with the same adjacency matrix have the same dimension and the same Forney indices and finally that for one-dimensional binary convolutional codes the adjacency matrix determines the code uniquely up to monomial equivalence.