Hyperoval code

Hyperoval code[1]

A projective code constructed using hyperovals in projective space.

\(q\)-ary sharp configuration— Codes based on hyperovals in \(PG_{2}(q)\) are \(q\)-ary sharp configurations [2; Table 12.1].
Projective two-weight code— Codes based on hyperovals in \(PG_{2}(q)\) with even \(q\) are projective two-weight codes [4,5][3; Exam. 19.2.1][6; Table 7.1].
Difference-set cyclic (DSC) code— Hyperoval difference sets yield DSC codes [7][8; Ch. 6].
Extended GRS code— Columns of parity-check matrices of triply extended RS codes consist of points of a hyperoval [9; Prop. 17.5].
Denniston code— Denniston codes for \(i=1\) are based on hyperovals in \(PG(2,2^a)\) [10].

Parents

Hyperoval code

Children

Columns of hexacode's generator matrix represent the six points of a hyperoval in the projective plane \(PG(2,4)\) [9; pg. 289][3; Exam. 19.2.1].

[1]: R. C. Bose (1947). Mathematical theory of the symmetrical factorial design. Sankhyā: The Indian Journal of Statistics, 107-166.
[2]: P. Boyvalenkov, D. Danev, "Linear programming bounds." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[3]: A. E. Brouwer, "Two-weight Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[4]: N. Tzanakis and J. Wolfskill, “The diophantine equation x2 = 4qa2 + 4q + 1, with an application to coding theory”, Journal of Number Theory 26, 96 (1987) DOI
[5]: A. E. Brouwer and H. Van Maldeghem, Strongly Regular Graphs (Cambridge University Press, 2022) DOI
[6]: R. A. Games, “The packing problem for projective geometries over GF(3) with dimension greater than five”, Journal of Combinatorial Theory, Series A 35, 126 (1983) DOI
[7]: A. Maschietti, Designs, Codes and Cryptography 14, 89 (1998) DOI
[8]: C. Ding, Codes from Difference Sets (WORLD SCIENTIFIC, 2014) DOI
[9]: J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016) DOI
[10]: Rudolf Schürer and Wolfgang Ch. Schmid. “Denniston Codes.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. https://mint.sbg.ac.at/desc_CDenniston.html

Page edit log

“Hyperoval code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/hyperoval