Spatially Coupled MacKay-Neal Codes Achieve Capacity on BMS Channels at Fixed Degrees
Abstract
We establish two degree-dependent results for spatially coupled MacKay-Neal codes on binary-input memoryless symmetric channels. The ensembles use uniform random smoothing and shorten both variable types outside the active chain. For r=g=3, every integer ℓ≥4, and each channel of capacity greater than R=r/ℓ, there is a sequence of code realizations whose actual transmitted rate tends to R and whose average sum-product bit error tends to zero. For r=g=2 and each ℓ∈{3,4,5}, an interval of binary symmetric channels has capacity greater than R but retains positive transmitted-bit error under terminated density evolution as chain length grows relative to coupling width. Both results follow from the signs of the same density-valued potential at uncoupled fixed points. The degree-three proof combines analytic bounds for ℓ≥33 with exact interval certificates for 4≤ℓ≤32, followed by threshold saturation and a projection argument for the actual rate. The degree-two proof analytically constructs a fixed point with negative potential and controls both boundary contributions. We relate the potential and the degree-two bifurcation condition to earlier statistical-mechanical predictions. The finite certificates and verification software are available in a versioned supplement.
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BibTeX
@misc{kasai2026spatially,
title = {{Spatially Coupled MacKay-Neal Codes Achieve Capacity on BMS Channels at Fixed Degrees}},
author = {Kenta Kasai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15526},
eprint = {2609.15526},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15526},
}
RIS
TY - UNPB
TI - Spatially Coupled MacKay-Neal Codes Achieve Capacity on BMS Channels at Fixed Degrees
AU - Kenta Kasai
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15526
UR - https://arxiv.org/abs/2609.15526
AB - We establish two degree-dependent results for spatially coupled MacKay-Neal codes on binary-input memoryless symmetric channels. The ensembles use uniform random smoothing and shorten both variable types outside the active chain. For r=g=3, every integer ℓ≥4, and each channel of capacity greater than R=r/ℓ, there is a sequence of code realizations whose actual transmitted rate tends to R and whose average sum-product bit error tends to zero. For r=g=2 and each ℓ∈{3,4,5}, an interval of binary symmetric channels has capacity greater than R but retains positive transmitted-bit error under terminated density evolution as chain length grows relative to coupling width. Both results follow from the signs of the same density-valued potential at uncoupled fixed points. The degree-three proof combines analytic bounds for ℓ≥33 with exact interval certificates for 4≤ℓ≤32, followed by threshold saturation and a projection argument for the actual rate. The degree-two proof analytically constructs a fixed point with negative potential and controls both boundary contributions. We relate the potential and the degree-two bifurcation condition to earlier statistical-mechanical predictions. The finite certificates and verification software are available in a versioned supplement.
ER -