Rate-Adaptive Spatially Coupled MacKay-Neal Codes with Thresholds Close to Capacity

TL;DR

This work analyzes rate-adaptive MacKay-Neal codes where an outer nonlinear distribution matcher is concatenated with an inner protograph spatially coupled LDPC code. Using density evolution on an equivalent parallel-channel model and a bit-scrambling trick to handle nonlinearity, the authors quantify BP decoding thresholds across multiple channel pairings. They show that SC MN codes can operate within a small gap to capacity across the full rate range, with the $\text{(4,8)}$ ensemble achieving a maximum gap of about $0.15$ dB on biAWGN, and overall SC MN performance beating the corresponding block MN codes by roughly $1$ dB. The results highlight the potential of rate-adaptive, single-inner-SC MN constructions for near-capacity, rate-flexible communications, enabled by tuning the outer distribution matcher parameter $\omega$.

Abstract

We analyze by density evolution the asymptotic performance of rate-adaptive MacKay-Neal (MN) code ensembles, where the inner code is a protograph spatially coupled (SC) low-density parity-check code. By resorting to a suitably-defined parallel channel model, we compute belief propagation decoding thresholds, showing that SC MN code ensembles can perform within 0.15 dB from the binary-input additive white Gaussian noise capacity over the full [0,1] rate range.

Figures (6)

• Figure 1: $(3, 6)$ protograph SC LDPC code ensemble from Example \ref{['ex:base']} (left) and of its SC counterpart (right). Dark variable nodes denote punctured nodes.
• Figure 2: System model, where a SC MN code is used to communicate over the biAWGN channel (communication channel).
• Figure 3: Modified SC MN scheme, with the introduction of the bit scrambler.
• Figure 4: Equivalent parallel channel model.
• Figure 5: Thresholds regions for block MN code ensembles (solid red/blue lines) and the corresponding SC MN code ensembles (dashed red/blue lines), compared to the rate-$1/2$ limit for parallel channel models (solid black line): BEC -- BEC (left), biAWGN -- biAWGN (center), and BSC -- biAWGN (right).

Theorems & Definitions (1)

• Example 1