Gaussian-Elimination-Free BP-OSD Algorithm for Short 5G LDPC Codes

Abstract

In this paper, we propose a two-stage belief-propagation ordered statistics decoding (BP-OSD)-like algorithm, hereafter referred to as BP-LCGCD, tailored to 5G low-density parity-check (LDPC) codes. When standard BP decoding fails, a modified BP (mBP) with an iteration limit determined by the code’s girth and a normalization factor $\alpha $ is performed to generate reliable posterior log-likelihood ratios (LLRs) in the pre-processing stage. In the post-processing stage, a locally constrained guessing codeword decoding (LC-GCD) module uses these posterior LLRs from the mBP to efficiently produce candidate codewords without any online Gaussian elimination (GE), where numerous invalid test error patterns are skipped by the introduced local constraints. To predict the decoding performance, we derive an upper bound on FER and employ a saddlepoint method to approximate its dominant tail-probability term, resulting in a semi-analytical FER prediction. Furthermore, we propose to design the second-stage LC-GCD to match the non-uniform posterior reliability profile induced by the BP-like preprocessing stage. To this end, an offline quasi-reliable re-encoding basis is constructed for 5G LDPC codes, where protograph EXIT-chart analysis is used to identify highly reliable variable-node positions. Simulation results demonstrate that the proposed GE-free BP-LCGCD significantly outperforms standard BP, achieves FER closely approaching that of GE-based BP-OSD, and approaches the finite-length bound across multiple code rates for 5G LDPC codes, while requiring a significantly smaller number of re-encodings and eliminating the need for online GE.