Design of Spatially Coupled LDPC Codes Over GF $(q)$ for Windowed Decoding

In this paper, we study spatially coupled lowdensity parity-check (SC-LDPC) codes over finite fields GF(q), q ≥ 2, and develop design rules for q-ary SC-LDPC code ensembles based on their iterative belief propagation decoding thresholds, with particular emphasis on low-latency windowed decoding (WD). We consider transmission over both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (BIAWGNC) and present results for a variety of (J, K)-regular SC-LDPC code ensembles constructed over GF(q) using protographs. Thresholds are calculated using the protograph versions of q-ary density evolution (for the BEC) and the q-ary extrinsic information transfer analysis (for the BIAWGNC). We show that the WD of q-ary SC-LDPC codes provides significant threshold gains compared with corresponding (uncoupled) q-ary LDPC block code (LDPC-BC) ensembles when the window size W is large enough and that these gains increase as the finite-field size q = 2m increases. Moreover, we demonstrate that the new design rules provide WD thresholds that are close to capacity, even when both m and W are relatively small (thereby reducing decoding complexity and latency). The analysis further shows that, compared with standard flooding-schedule decoding, the WD of q-ary SC-LDPC code ensembles results in significant reductions in both the decoding complexity and the decoding latency and that these reductions increase as m increases. For the applications with a near-threshold performance requirement and a constraint on decoding latency, we show that using q-ary SC-LDPC code ensembles, with moderate q > 2, instead of their binary counterparts results in reduced decoding complexity.