## Description

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.

## Rate

## Encoding

## Decoding

## Notes

## Parents

## Children

## Cousins

- Tensor-product code — Tensor products of random LDPC codes are robustly testable [39,40].
- Low-density generator-matrix (LDGM) code — The dual of an LDPC code has a sparse generator matrix and is called an LDGM code.
- Random code — LDPC codes are often constructed non-determinisitically.
- Hamiltonian-based code — There are relations between LDPC codes and statistical mechanical models of spin glasses [25,26,41].
- Tornado code — Tornado codes are similar to LDPC codes, but they use a highly irregular weight distribution for the underlying graphs [42].
- Low-rank parity-check (LRPC) code — LRPC codes are rank-metric analogues of LDPC codes [43].
- DNA storage code — LDPC codes are potentially relevant for DNA storage [44].
- LDPC convolutional code (LDPC-CC) — LDPC-CCs are convolutional analogues of LDPC codes.
- Lechner-Hauke-Zoller (LHZ) code — The LHZ code is an LDPC c-q code designed to convert the long-range interactions of a quantum annealer into local constraints.
- Concatenated cat code — Cat codes have been concatenated with LDPC codes (treated as qubit stabilizer codes) [45].
- Quantum LDPC (QLDPC) code
- Spacetime circuit code — There is an equivalence between Clifford circuits and LDPC codes with bit-check symmetry [46].
- EA QLDPC code — There exist necessary and sufficient conditions for an EA QLDPC code consuming \(e=1\) ebit that is obtainable from a pair of LDPC codes [47].

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## 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/edit/main/codes/classical/bits/tanner/ldpc.yml.