On the MacWilliams Identity for Convolutional Codes

TL;DR

This paper explores a conjecture relating the adjacency matrices of convolutional codes and their duals, providing explicit formulas and proofs for specific cases, advancing understanding of weight distributions in convolutional coding theory.

Contribution

It formulates a MacWilliams Identity Conjecture for convolutional codes and proves it for codes with low Forney indices, offering explicit transformation formulas.

Findings

Proves the conjecture for codes with Forney indices ≤ 1

Provides an explicit formula involving the MacWilliams matrix

Supports the conjecture with numerous examples in the general case

Abstract

The adjacency matrix associated with a convolutional code collects in a detailed manner information about the weight distribution of the code. A MacWilliams Identity Conjecture, stating that the adjacency matrix of a code fully determines the adjacency matrix of the dual code, will be formulated, and an explicit formula for the transformation will be stated. The formula involves the MacWilliams matrix known from complete weight enumerators of block codes. The conjecture will be proven for the class of convolutional codes where either the code itself or its dual does not have Forney indices bigger than one. For the general case the conjecture is backed up by many examples, and a weaker version will be established.