Spatially Coupled MacKay-Neal Codes on Symmetric Markov Noise: A Reduction to BMS Channels

2026-09-22

AI Summary

This study addresses the unresolved capacity criterion for fixed-degree spatially coupled codes over symmetric Markov noise channels. To establish this criterion, the authors reduce the channel with memory to a binary memoryless symmetric (BMS) channel via conditional entropy correction. Furthermore, they introduce a two-state posterior contraction method to refine derivative bounds, thereby enabling threshold saturation analysis. The theoretical derivation is accomplished by integrating BCJR and sum-product decoding, leveraging a Gilbert–Elliott channel to binary erasure channel analogy, and employing interval certificate techniques. Ultimately, this work proves the existence of code sequences that approach channel capacity and rigorously demonstrates that the average information bit error rate vanishes asymptotically, achieving a complete analytical proof with exact verification.

Abstract

We prove a capacity criterion for fixed-degree spatially coupled MacKay--Neal codes on additive symmetric Markov noise. The noise bit flips with probability $p$, so the channel has one parameter and capacity $1-h_2(p)$. For every integer $\ell\geq4$ with $3/\ell<1-h_2(p)$, matched finite-window BCJR and sum-product decoding admit code sequences, possibly depending on $p$, with actual rate tending to $3/\ell$ and vanishing average information-bit error. The proof parallels the GEC-to-BEC argument: a conditional-entropy correction transfers BMS fixed-point positivity to the channel with memory. The correction also covers a capacity deficit of the effective BMS channel. A two-state posterior contraction supplies the additional derivative bound needed for threshold saturation. The result uses the cited BMS positivity theorem, with an analytic proof for $\ell\geq33$ and exact interval certificates for $4\leq\ell\leq32$.

Problem

• Spatially Coupled MacKay-Neal Codes
• Symmetric Markov Noise
• Capacity Criterion
• BMS Channels
• Threshold Saturation

Innovation

• Spatially Coupled MacKay-Neal Codes
• Symmetric Markov Noise
• Threshold Saturation
• Conditional-Entropy Correction
• BCJR Decoding