The Mutual information of LDGM codes
Jan van den Brand, Nor Jaafari
TL;DR
This paper establishes tight bounds on the mutual information in noisy LDGM code reconstruction, confirming a long-standing conjecture and extending prior results to more general code ensembles.
Contribution
It provides the first proof of the conjectured mutual information formula for LDGM codes with arbitrary degree distributions, extending previous concentration results.
Findings
Matching upper and lower bounds on mutual information
Confirmation of Montanari's conjecture for LDGM codes
Extension of concentration results to odd check degrees
Abstract
We provide matching upper and lower bounds on the mutual information in noisy reconstruction of parity check codes and thereby prove a long-standing conjecture by Montanari [IEEE Transactions on Information Theory 2005]. Besides extending a prior concentration result of Abbe and Montanari [Theory of Computing 2015] to the case of odd check degrees, we precisely determine the conjectured formula for code ensembles of arbitrary degree distribution, thus capturing a broad class of capacity approaching codes.
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Taxonomy
TopicsError Correcting Code Techniques · Coding theory and cryptography · Cryptography and Data Security