Codes on graphs: Duality and MacWilliams identities
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.
Cite
@article{arxiv.0911.5508,
title = {Codes on graphs: Duality and MacWilliams identities},
author = {G. David Forney},
journal= {arXiv preprint arXiv:0911.5508},
year = {2010}
}
Comments
30 pages, 20 figures. Accepted for IEEE Transactions on Information Theory. Preliminary versions presented at IEEE Information Theory Workshop, Taormina, Italy, October 2009 and 2010 IEEE International Symposium on Information Theory, Austin, TX, June 2010