LDPC Code Density Evolution in the Error Floor Region

Abstract—This short paper explores density evolution (DE) forlow-density parity-check (LDPC) codes at signal-to-noise-ratios(SNRs) that are significantly above the decoding threshold. Thefocus is on the additive white Gaussian noise channel and LDPCcodes in which the variable nodes have regular degree.Prior work, using DE, produced results in the error floorregion which were asymptotic in the belief-propagation decoder’slog-likelihood ratio (LLR) values. We develop expressions whichclosely approximate the LLR growth behavior at moderate LLRmagnitudes. We then produce bounds on the mean extrinsiccheck-node LLR values required, as a function of SNR, such thatthe growth rate of the LLRs exceeds that of a particular trappingset’s internal LLRs such that its error floor contribution may beeliminated. We find that our predictions for the mean LLRs tobe accurate in the error floor region, but the predictions for theLLR variance to be lacking beyond several initial iterations.Index Terms—Low-density parity-check (LDPC) code, beliefpropagation (BP), sum-product algorithm (SPA) decoding, den-sity evolution, error floor, Margulis code, trapping set.