Incidence-matrix projective code

Incidence-matrix projective code[1–3]

Description

Code whose generator matrix is the incidence matrix of points and hyperplanes in a projective space. Has been generalized to incidence matrices of other structures [4,5][6; Sec. 14.4]. Columns of a code's parity-check matrix can similarly correspond to an incidence matrix.

Cousins

\([7,4,3]\) Hamming code— The \([7,4,3]\) Hamming code parity-check matrix corresponds to points in the Fano plane \(PG_2(2)\) [7; Exam. 21.4.2].
Finite-geometry LDPC (FG-LDPC) code— The parity-check matrix of a PG-LDPC code is the incidence matrix of points and hyperplanes in a projective space.
\(q\)-ary Hamming code— Columns of a Hamming parity-check matrix correspond to one-dimensional subspaces of \(GF(q)^n\).

Member of code lists

Primary Hierarchy

Classical Domain

Galois-field Kingdom

\(q\)-ary codeECC

Additive \(q\)-ary codeECC

Linear \(q\)-ary codeECC

Projective geometry code

Parents

Projective geometry code

Incidence-matrix projective code

Children

\(q\)-ary simplex code

Columns of a simplex code's generator matrix correspond to one-dimensional subspaces of \(GF(q)^n\).

References

[1]: E. Prange, The use of coset equivalene in the analysis and decoding of group codes. AIR FORCE CAMBRIDGE RESEARCH LABS HANSCOM AFB MA, 1959.
[2]: L. Rudolph, “A class of majority logic decodable codes (Corresp.)”, IEEE Transactions on Information Theory 13, 305 (1967) DOI
[3]: E. Prange, "Some cyclic error-correcting codes with simple decoding algorithms." AFCRC-TN-58-156 (1985).
[4]: B. Bagchi and S. P. Inamdar, “Projective Geometric Codes”, Journal of Combinatorial Theory, Series A 99, 128 (2002) DOI
[5]: M. Lavrauw, L. Storme, and G. Van de Voorde (2010). Linear codes from projective spaces. In A. Bruen & D. Wehlau (Eds.), Contemporary Mathematics (Vol. 523, pp. 185–202). Providence, RI, USA: American Mathematical Society (AMS).
[6]: L. Storme, "Coding Theory and Galois Geometries." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[7]: C. A. Kelley, "Codes over Graphs." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI

Page edit log

Victor V. Albert (2022-08-10) — most recent

Cite as:

“Incidence-matrix projective code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/incidence_matrix

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/projective/incidence_matrix.yml.