## 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

## Rate

## Encoding

## Decoding

## Realizations

## Notes

## Parent

## Child

## 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

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- 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