## Description

CSS code constructed from linear binary codes whose parity-check or generator matrices are incidence matrices of points, hyperplanes, or other structures in finite geometries. These codes can be interpreted as quantum versions of FG-LDPC codes, but some of them [2,3] are not strictly QLDPC.

## Parents

## Child

- \([[7,1,3]]\) Steane code — The Steane code is the \(m=1\) member of the \([[2^{2m}+2^{m}+1,1,>2^{m}]]\) PG-QLDPC code family that is constructed from codes corresponding to lines and affine charts in \(PG(2,2^m)\) via the CSS construction [3; Def. 4.9].

## Cousins

- Finite-geometry LDPC (FG-LDPC) code — Quantum versions of PG-LDPC and EG-LDPC codes can be constructed via the CSS construction [1–3].
- Projective geometry code — PG-QLDPC codes are constructed from linear binary codes whose parity-check or generator matrices are incidence matrices of structures in finite geometries.
- Generalized homological-product qubit CSS code — Iterative tensor products of PG-QLDPC codes yield codes whose stabilizer-generator weights scale almost logarithmically with \(n\) [3].

## References

- [1]
- 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
- [2]
- J. Farinholt, “Quantum LDPC Codes Constructed from Point-Line Subsets of the Finite Projective Plane”, (2012) arXiv:1207.0732
- [3]
- B. Audoux and A. Couvreur, “On tensor products of CSS Codes”, (2018) arXiv:1512.07081

## Page edit log

- Victor V. Albert (2024-05-03) — most recent

## Cite as:

“Finite-geometry (FG) QLDPC code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/pg_qldpc