Generation, Encoding, and Decoding Process of LDPC Parity-Check Matrix

In this article, we will explore in detail the generation of the parity-check matrix, encoding, and decoding processes for LDPC codes. LDPC codes are a type of linear block code that operates based on matrix concepts, where encoding and decoding are implemented through matrix operations. The generation of the parity-check matrix is a critical step in LDPC codes, typically implemented using algorithms like the Gallager construction, progressive edge growth (PEG), or quasi-cyclic methods. This process connects information bits with parity bits to form a specific sparse matrix structure, providing the fundamental basis for both encoding and decoding operations. The encoding process involves transforming input information bits into LDPC codewords. This is typically achieved through matrix multiplication using a generator matrix derived from the parity-check matrix, or more efficiently via direct encoding algorithms that leverage the sparse structure of the matrix. Key implementation approaches include using Gaussian elimination to transform the parity-check matrix into systematic form or employing back-substitution methods for lower complexity. The decoding process converts received LDPC codes back to original information bits. This is commonly implemented using iterative message-passing algorithms like the Sum-Product Algorithm (SPA) or the Min-Sum Algorithm (MSA), which operate on the Tanner graph representation of the parity-check matrix. These algorithms work by passing probabilistic messages between variable nodes and check nodes, with implementations typically involving log-likelihood ratio (LLR) calculations and iterative updates until convergence criteria are met. In this article, we will provide detailed explanations of the parity-check matrix generation methods, encoding procedures, and decoding processes for LDPC codes, including relevant algorithm implementations and key function descriptions to help readers better understand and apply LDPC codes in practical systems.