# Finite-geometry LDPC (FG-LDPC) code[1]

## Description

LDPC code whose parity-check matrix is the incidence matrix of points and hyperplanes in either a Euclidean or a projective geometry. Such codes are called Euclidean-geometry LDPC (EG-LDPC) and projective-geometry LDPC (PG-LDPC), respectively. Such constructions have been generalized to incidence matrices of hyperplanes of different dimensions [2].

## Parents

## Cousins

- Quasi-cyclic LDPC (QC-LDPC) code — Many FG-LDPC codes can be put into quasi-cyclic form [1,2][3; pg. 286].
- Incidence-matrix projective code — The parity-check matrix of a PG-LDPC code is the incidence matrix of points and hyperplanes in a projective space.
- Quantum low-density parity-check (QLDPC) code — Quantum versions of EG LDPC codes can be constructed via the CSS construction [4].

## References

- [1]
- Y. Kou, S. Lin, and M. P. C. Fossorier, “Low-density parity-check codes based on finite geometries: a rediscovery and new results”, IEEE Transactions on Information Theory 47, 2711 (2001) DOI
- [2]
- Heng Tang et al., “Codes on finite geometries”, IEEE Transactions on Information Theory 51, 572 (2005) DOI
- [3]
- Latin Squares and Their Applications (Elsevier, 2015) DOI
- [4]
- S. A. Aly, “A Class of Quantum LDPC Codes Constructed From Finite Geometries”, IEEE GLOBECOM 2008 - 2008 IEEE Global Telecommunications Conference (2008) arXiv:0712.4115 DOI

## Page edit log

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

## Cite as:

“Finite-geometry LDPC (FG-LDPC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/pg_ldpc