# 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.
- Generalized RM (GRM) code — Some EG-LDPC codes are duals of subfield subcodes of GRM codes [4; pg. 448].
- Asymmetric quantum code — EG-LDPC codes can be used to construct asymmetric CSS codes [5; Lemma 4.1].
- Finite-geometry (FG) QLDPC code — Quantum versions of PG-LDPC and EG-LDPC codes can be constructed via the CSS construction [6–8].
- Abelian LP code — FG-LDPC codes can be used to construct Abelian LP codes [9].

## 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”, (2015) DOI
- [4]
- R. E. Blahut, Algebraic Codes for Data Transmission (Cambridge University Press, 2003) DOI
- [5]
- P. K. Sarvepalli, A. Klappenecker, and M. Rötteler, “Asymmetric quantum codes: constructions, bounds and performance”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, 1645 (2009) DOI
- [6]
- 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
- [7]
- J. Farinholt, “Quantum LDPC Codes Constructed from Point-Line Subsets of the Finite Projective Plane”, (2012) arXiv:1207.0732
- [8]
- B. Audoux and A. Couvreur, “On tensor products of CSS Codes”, (2018) arXiv:1512.07081
- [9]
- S. Miao et al., “A Joint Code and Belief Propagation Decoder Design for Quantum LDPC Codes”, (2024) arXiv:2401.06874

## 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