Incidence-matrix projective code[1–3] 

Description

Code whose generator matrix is the incidence matrix of points and hyperplanes of a projective spaces. 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.

Parent

Child

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

Cousins

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
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Zoo Code ID: incidence_matrix

Cite as:
“Incidence-matrix projective code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/incidence_matrix
BibTeX:
@incollection{eczoo_incidence_matrix,
  title={Incidence-matrix projective code},
  booktitle={The Error Correction Zoo},
  year={2022},
  editor={Albert, Victor V. and Faist, Philippe},
  url={https://errorcorrectionzoo.org/c/incidence_matrix}
}
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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/tree/main/codes/classical/q-ary_digits/projective/incidence_matrix.yml.