# \(q\)-ary LDPC code[1]

## Description

A \(q\)-ary linear code with a sparse parity-check matrix. Alternatively, a member of an infinite family of \([n,k,d]_q\) 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 \(n\to\infty\).

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 a nonzero field element (error). Despite the fact that there is more than one nonzero outcome, \(q>2g\)-ary linear codes with sparse parity-check matrices are also called LDPC codes.

## Protection

## Parents

- Tanner code — \(q\)-ary LDPC codes are \(q\)-ary Tanner codes on sparse bipartite graphs whose constraint nodes represent \(q\)-ary parity-check codes.
- Locally recoverable code (LRC) — LDPC codes are linear LRCs whose locality is the maximum number of nonzero entries in a row of the parity-check matrix [5].

## Children

## Cousin

- \(q\)-ary LDGM code — The dual of a \(q\)-ary LDPC code has a sparse generator matrix and is called a \(q\)-ary LDGM code.

## References

- [1]
- M. C. Davey and D. J. C. MacKay, “Low density parity check codes over GF(q)”, 1998 Information Theory Workshop (Cat. No.98EX131) DOI
- [2]
- X.-Yu. Hu and E. Eleftheriou, “Binary representation of cycle Tanner-graph GF(2/sup b/) codes”, 2004 IEEE International Conference on Communications (IEEE Cat. No.04CH37577) (2004) DOI
- [3]
- R.-H. Peng and R.-R. Chen, “Design of Nonbinary Quasi-Cyclic LDPC Cycle Codes”, 2007 IEEE Information Theory Workshop (2007) DOI
- [4]
- C. Poulliat, M. Fossorier, and D. Declercq, “Design of regular (2,d/sub c/)-LDPC codes over GF(q) using their binary images”, IEEE Transactions on Communications 56, 1626 (2008) DOI
- [5]
- L. Golowich and V. Guruswami, “Quantum Locally Recoverable Codes”, (2023) arXiv:2311.08653

## Page edit log

- Victor V. Albert (2023-05-04) — most recent

## Cite as:

“\(q\)-ary LDPC code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/q-ary_ldpc