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

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 $\ell \ge 4$ 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 $\ell \ge 33$ and exact interval certificates for $4\le \ell \le 32$.

arXiv comment15 pages