Codes on graphs: Duality and MacWilliams identities

TL;DR

This paper introduces a framework using partition functions of normal factor graphs to establish duality and MacWilliams identities, generalizing previous results for various codes on graphs with a unified Fourier transform perspective.

Contribution

It develops a general framework linking normal graph duality with Fourier transforms and derives new MacWilliams identities for a broad class of codes, including convolutional and tail-biting codes.

Findings

Partition functions of dual graphs form a Fourier transform pair.

MacWilliams identities are extended to general group and linear codes on graphs.

New identities are derived for terminated and tail-biting convolutional codes.

Abstract

A conceptual framework involving partition functions of normal factor graphs is introduced, paralleling a similar recent development by Al-Bashabsheh and Mao. The partition functions of dual normal factor graphs are shown to be a Fourier transform pair, whether or not the graphs have cycles. The original normal graph duality theorem follows as a corollary. Within this framework, MacWilliams identities are found for various local and global weight generating functions of general group or linear codes on graphs; this generalizes and provides a concise proof of the MacWilliams identity for linear time-invariant convolutional codes that was recently found by Gluesing-Luerssen and Schneider. Further MacWilliams identities are developed for terminated convolutional codes, particularly for tail-biting codes, similar to those studied recently by Bocharova, Hug, Johannesson and Kudryashov.