Spatially coupled MacKay-Neal codes achieve channel capacity at fixed degrees
Spatially Coupled MacKay-Neal Codes Achieve Capacity on BMS Channels at Fixed Degrees
Information Theory
Summary
Communicating information over noisy channels without errors is a fundamental challenge. This paper shows that a special type of error-correcting code, called spatially coupled MacKay-Neal codes, can reliably send data at rates close to the channel's maximum capacity when the coding structure has fixed, small complexity. The authors prove this specifically for certain parameters and channels, while also identifying cases where errors persist. Their approach involves deep mathematical analysis and provides exact numerical verification.
What this means in practice
- •For communication system engineers: Design error-correcting codes with fixed complexity that approach theoretical capacity limits on symmetric noisy channels for improved data transmission.
- •For digital hardware designers: Implement spatially coupled coding schemes with predictable performance for reliable data transfer in constrained-complexity hardware systems.
Authors
Kenta Kasai
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 $\ell\geq4$, and each channel of capacity greater than $R=r/\ell$, 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 $\ell\in\{3,4,5\}$, 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 $\ell\geq33$ with exact interval certificates for $4\leq\ell\leq32$, 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.