Low-density parity-check (LDPC) code[1,2] 

Also known as Sparse graph code.


A binary linear code with a sparse parity-check matrix. Alternatively, a member of an infinite family of \([n,k,d]\) codes for which the number of nonzero entries in each row and column of the parity-check matrix are both bounded above by a constant as \(n\to\infty\).

An LDPC code is \((j,k)\)-regular if the parity-check matrix has a fixed number of \(j\) nonzero entries in each row and \(k\) entries in each column; otherwise, the LDPC code is irregular. Irregular LDPC codes are characterized by the fractions \(v_j\) of columns of weight \(j\) as well as likewise fractions \(h_j\) for rows of weight \(j\); these are collectively called the degree distribution.

A parity check is performed by taking the inner product of a row of the parity-check matrix with a codeword that has been affected by a noise channel. A parity check yields either zero (no error) or one (error).

In alternative conventions (not used here), LDPC codes are referred to as simple LDPC codes, as opposed to generalized LDPC codes, alternatively named as Tanner codes.


Some LDPC codes achieve the Shannon capacity of the binary symmetric channel under maximum-likelihood decoding [1,3,4]. Other LDPC codes achieve capacity for smaller block lengths under belief-propagation decoding [5]. Random LDPC codes achieve list-decoding capacity [6].


Almost linear-time encoder based on transforming the parity-check matrix into upper triangular form [7].


Message-passing algorithm called belief propagation (BP) [2,8,9] (see also [1012]).Soft-decision Sum-Product Algorithm (SPA) [2,10,13] and its simplification the Min-Sum Algorithm (MSA) [14].Linear programming [1517].Iterative LDPC decoders can get stuck at stopping sets of their Tanner graphs [18], with decoder performance improving with the size of the smallest stopping set; see [19; Sec. 21.3.1] for more details. The smallest stopping set size can reach the minimum distance of the code [20].Ensembles of random LDPC codes under iterative decoders are subject to the concentration theorem [10,21]; see [19; Thm. 21.7.1] for the case of the BEC.Reinforcement learning [22].


The potential of LDPC codes was noted by Margulis [23], but realized by the broader community [3,24] much later after their discovery by Gallager [1,2].See book [25] and reviews [26,27] for an introduction to LDPC codes, belief-propagation decoding, and connections to statistical mechanics. Other introductory references include Refs. [2831] as well as a review of LDPC codes circa 2005 [32].See Kaiserslautern database [33] for explicit representatives of several classes of LDPC codes, including \(q\)-ary, WiMAX, multi-edge, and spatially-coupled.See pretty-good-codes database [34] for explicit representatives and benchmarking.See Encyclopedia of sparse graph codes for explicit representatives [35]LDPC codes have been considered for quantum key distribution [36].Codes have been benchmarked using AFF3CT toolbox [37].





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“Low-density parity-check (LDPC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/ldpc
@incollection{eczoo_ldpc, title={Low-density parity-check (LDPC) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/ldpc} }
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