MacWilliams Identities for Codes on Graphs
G. David Forney Jr
TL;DR
This paper proves and generalizes MacWilliams identities for codes on graphs, providing concise proofs and extending the results to arbitrary group codes, enhancing understanding of code duality and decoding algorithms.
Contribution
It offers a concise proof of MacWilliams identities for group codes on graphs and generalizes these identities beyond linear convolutional codes.
Findings
Proved MacWilliams identities for group codes on graphs
Generalized identities to arbitrary group codes
Provided a transparent proof of the dual sum-product update rule
Abstract
The MacWilliams identity for linear time-invariant convolutional codes that has recently been found by Gluesing-Luerssen and Schneider is proved concisely, and generalized to arbitrary group codes on graphs. A similar development yields a short, transparent proof of the dual sum-product update rule.
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Taxonomy
TopicsCooperative Communication and Network Coding · Coding theory and cryptography · Error Correcting Code Techniques