Low-density parity-check (LDPC) code

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

Description

Also known as Gallager codes. A \(q\)-ary linear code with a sparse parity-check matrix. More precisely, 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 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. The dual of an LDPC code has a sparse generator matrix and is called an LDGM code.

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) for binary codes, while yielding zero (no error) or a nonzero field element (error) for \(q\)-ary codes. Despite the fact that there is more than one nonzero outcome, \(q\)-ary linear codes with sparse parity-check matrices are also called LDPC codes.

Protection

With high probability, random LDPC codes have constant distance [2].

Rate

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

Encoding

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

Decoding

Message-passing algorithm called belief propagation (BP) [2][8].Linear programming [9].Reinforcement learning [10].

Realizations

5G NR cellular communication for the traffic channel [11].WiMAX (IEEE 802.16e) [12].Satellite transmission of digital television [13].Quantum key distribution [14].

Notes

See Kaiserslautern database [15] for explicit representatives of several classes of LDPC codes, including \(q\)-ary, WiMAX, multi-edge, and spatially-coupled.See pretty-good-codes database [16] for explicit representatives and benchmarking.See Ref. [17] for a review of LDPC codes circa 2005.Codes have been benchmarked using AFF3CT toolbox [18].See book [19] for an introduction to LDPC codes, belief-propagation decoding, and connections to statistical mechanics.

Parent

Linear \(q\)-ary code

Child

Tornado code

Cousins

Low-density generator-matrix (LDGM) code — LDPC and LDGM codes are dual to each other.
Tensor-product code — Tensor products of random LDPC codes are robustly testable [20][21].
Tornado code — Tornado codes are similar to LDPC codes, but they use a highly irregular weight distribution for the underlying graphs [22].
Low-rank parity-check (LRPC) code — LRPC codes are rank-metric analogues of LDPC codes [23].
Quantum low-density parity-check (QLDPC) code

References

[1]: R. Gallager, “Low-density parity-check codes”, IEEE Transactions on Information Theory 8, 21 (1962) DOI
[2]: R. Gallagher, Low-density parity check codes. 1963. PhD thesis, MIT Cambridge, MA.
[3]: D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices”, IEEE Transactions on Information Theory 45, 399 (1999) DOI
[4]: V. Guruswami, “Iterative Decoding of Low-Density Parity Check Codes (A Survey)”, (2006) arXiv:cs/0610022
[5]: S. Kudekar, T. Richardson, and R. Urbanke, “Spatially Coupled Ensembles Universally Achieve Capacity under Belief Propagation”, (2012) arXiv:1201.2999
[6]: J. Mosheiff et al., “LDPC Codes Achieve List Decoding Capacity”, (2021) arXiv:1909.06430
[7]: T. J. Richardson and R. L. Urbanke, “Efficient encoding of low-density parity-check codes”, IEEE Transactions on Information Theory 47, 638 (2001) DOI
[8]: S. Lin and D. J. Costello, Error Control Coding, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 2004.
[9]: J. Feldman, “LP Decoding”, Encyclopedia of Algorithms 1177 (2016) DOI
[10]: S. Habib, A. Beemer, and J. Kliewer, “RELDEC: Reinforcement Learning-Based Decoding of Moderate Length LDPC Codes”, (2021) arXiv:2112.13934
[11]: M. V. Patil, S. Pawar, and Z. Saquib, “Coding Techniques for 5G Networks: A Review”, 2020 3rd International Conference on Communication System, Computing and IT Applications (CSCITA) (2020) DOI
[12]: LDPC coding for OFDMA PHY. 802.16REVe Sponsor Ballot Recirculation comment, July 2004. IEEE C802.16e04/141r2
[13]: R. Purnamasari, H. Wijanto, and I. Hidayat, “Design and implementation of LDPC(Low Density Parity Check) coding technique on FPGA (Field Programmable Gate Array) for DVB-S2 (Digital Video Broadcasting-Satellite)”, 2014 IEEE International Conference on Aerospace Electronics and Remote Sensing Technology (2014) DOI
[14]: N. Borisov, I. Petrov, and A. Tayduganov, “Asymmetric Adaptive LDPC-Based Information Reconciliation for Industrial Quantum Key Distribution”, Entropy 25, 31 (2022) arXiv:2212.01121 DOI
[15]: Michael Helmling, Stefan Scholl, Florian Gensheimer, Tobias Dietz, Kira Kraft, Stefan Ruzika, and Norbert Wehn. Database of Channel Codes and ML Simulation Results. URL, 2022.
[16]: G. Liva, F. Steiner. “pretty-good-codes.org: Online library of good channel codes”, URL: http://pretty-good-codes.org/
[17]: A. Shokrollahi, “LDPC Codes: An Introduction”, Coding, Cryptography and Combinatorics 85 (2004) DOI
[18]: A. Cassagne et al., “AFF3CT: A Fast Forward Error Correction Toolbox!”, SoftwareX 10, 100345 (2019) DOI
[19]: M. Mézard and A. Montanari, Information, Physics, and Computation (Oxford University PressOxford, 2009) DOI
[20]: I. Dinur, M. Sudan, and A. Wigderson, “Robust Local Testability of Tensor Products of LDPC Codes”, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques 304 (2006) DOI
[21]: E. Ben-Sasson and M. Viderman, “Tensor Products of Weakly Smooth Codes Are Robust”, Lecture Notes in Computer Science 290 (2008) DOI
[22]: A. Shokrollahi, “Raptor codes”, IEEE Transactions on Information Theory 52, 2551 (2006) DOI
[23]: Gaborit, P., Murat, G., Ruatta, O., & Zemor, G. (2013, April). Low rank parity check codes and their application to cryptography. In Proceedings of the Workshop on Coding and Cryptography WCC (Vol. 2013).

Page edit log

Victor V. Albert (2022-08-17) — most recent
Victor V. Albert (2022-04-25)
Armin Gerami (2022-04-23)

Cite as:

“Low-density parity-check (LDPC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/ldpc

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/q-ary_digits/ldpc/ldpc.yml.