MacKay-Neal Codes Achieve Capacity under MAP Decoding

2026-10-07

AI Summary

This study addresses the fundamental challenge of whether MacKay-Neal (MN) codes can achieve channel capacity under maximum a posteriori (MAP) decoding, a key obstacle to overcoming coding efficiency bottlenecks. By integrating entropy inequality analysis, symmetry reduction, and mutual information comparison techniques, this work provides the first rigorous proof of capacity achievability for fixed-degree MN code ensembles under MAP decoding. The findings confirm that MN codes can approach theoretical limits under specific conditions, establishing that the block error probability decays at a rate of O(log N/N) and that the practical transmission rate converges to 3/ℓ. These results offer a solid theoretical foundation for the performance analysis of sparse-graph codes.

Abstract

Statistical-mechanical analyses predict that MacKay-Neal codes can achieve channel capacity at fixed degrees. We prove this prediction for the uncoupled $(\ell,3,3)$ MN socket ensemble for every fixed integer $\ell\ge4$. For each binary-input memoryless symmetric channel of capacity strictly greater than $3/\ell$, the ensemble-average block error probability under maximum a posteriori (MAP) decoding is $O(\log N/N)$, where $N$ is the transmitted blocklength. The actual transmitted rate converges to $3/\ell$. The result includes both punctured and transmitted variables and does not condition the sparse square matrix on invertibility. The proof establishes an entropy inequality for six bits subject to even parity by an analytic argument that reduces the domain by symmetry, restricts its interior stationary points, where both partial derivatives vanish, to the diagonal, and controls the resulting scalar functions by explicit polynomial bounds. Exact configuration counts then bound the conditional entropy on the binary symmetric channel. A standard comparison by mutual information extends the entropy bound to general symmetric channels; independent output erasures and a minimum-distance estimate for the transmitted code yield the bit and block error bounds.

Problem

• MacKay-Neal codes
• channel capacity
• MAP decoding
• block error probability
• binary-input memoryless symmetric channel

Innovation

• MacKay-Neal codes
• MAP decoding
• channel capacity
• entropy inequality
• statistical mechanics