LDPCCodes

LDPCCodes are a class of linear error-correcting codes also known as LDPC codes. They are defined by a sparse parity-check matrix H with relatively few nonzero entries, which enables efficient iterative decoding. Codewords satisfy H times x equals zero over a binary field, and the parameters n (block length) and k (message length) determine the code rate R = k/n.

LDPCCodes are typically represented by a bipartite Tanner graph with variable nodes (code bits) and check nodes (parity checks). Decoding proceeds by passing probabilistic messages along edges in an iterative fashion, commonly via the belief propagation or sum-product algorithm. Performance approaches the Shannon limit for long block lengths, with decoding complexity roughly proportional to the sparsity of H.

History and varieties: First proposed by Gallager in 1962, their practical revival in the 1990s led to many optimized irregular and protograph-based constructions. Irregular LDPC codes use variable-degree distributions to improve thresholds; protograph LDPC codes offer structured designs suited for hardware; spatially coupled LDPC codes provide excellent finite-length performance and robust thresholds.

Applications: LDPCCodes are widely used in modern communications and storage systems. They are standardized in Wi-Fi (IEEE 802.11n/ac/ax), DVB-S2, and are a key component in 5G NR, as well as in various optical and data-storage standards.

Design considerations: choosing degree distributions, block length, rate, puncturing and shortening, and decoding schedules affect performance and complexity. Implementations may use hardware-friendly architectures to maximize throughput and minimize latency.