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

## Description

Also known as Gallager codes. Family of \([n,k,d]\) linear 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 \(n\to\infty\). In other words, the parity-check matrix of an LDPC code is sparse.

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.

## Realizations

## Parent

## Cousins

- Linear binary code — Binary LDPC codes are a subset of binary linear codes.
- Linear \(q\)-ary code — \(q\)-ary LDPC codes are a subset of \(q\)-ary linear codes.
- Quantum low-density parity-check (QLDPC) code
- Tornado code — Tornado codes are similar to LDPC codes, but they use a highly irregular weight distribution for the underlying graphs [5].

## Zoo code information

## References

- [1]
- R. Gallager, “Low-density parity-check codes”, IEEE Transactions on Information Theory 8, 21 (1962). DOI
- [2]
- 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
- [3]
- LDPC coding for OFDMA PHY. 802.16REVe Sponsor Ballot Recirculation comment, July 2004. IEEE C802.16e04/141r2
- [4]
- 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
- [5]
- A. Shokrollahi, “Raptor codes”, IEEE Transactions on Information Theory 52, 2551 (2006). DOI

## 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/properties/ldpc.yml.