## Description

Binary linear code with a sparse generator 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 generator matrix are both bounded by a constant as \(n\to\infty\). The dual of an LDGM code has a sparse parity-check matrix and is called an LDPC code.

## Rate

Certain LDGM codes come close to achieving Shannon capacity [1].

## Parents

## Child

## Cousins

- Single parity-check (SPC) code — Concatenated SPCs are LDGM [2].
- Irregular repeat-accumulate (IRA) code — IRA codes replace the outer 1-in-3 repetition encoding step in RA codes with an LDGM code.
- Low-density parity-check (LDPC) code — The dual of an LDPC code has a sparse generator matrix and is called an LDGM code.
- Hsu-Anastasopoulos LDPC (HA-LDPC) code — HA-LDPC codes are a concatenation of an LDPC and an LDGM code.
- MacKay-Neal LDPC (MN-LDPC) code — \((l,r,1\))-MN-LDPC codes are LDGM [3].

## References

- [1]
- J. Garcia-Frias and Wei Zhong, “Approaching Shannon performance by iterative decoding of linear codes with low-density generator matrix”, IEEE Communications Letters 7, 266 (2003) DOI
- [2]
- T. R. Oenning and Jaekyun Moon, “A low-density generator matrix interpretation of parallel concatenated single bit parity codes”, IEEE Transactions on Magnetics 37, 737 (2001) DOI
- [3]
- K. KASAI and K. SAKANIWA, “Spatially-Coupled MacKay-Neal Codes and Hsu-Anastasopoulos Codes”, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E94-A, 2161 (2011) arXiv:1102.4612 DOI

## Page edit log

- Victor V. Albert (2022-08-15) — most recent

## Cite as:

“Low-density generator-matrix (LDGM) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/ldgm

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/bits/tanner/ldgm.yml.