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Denniston code[1]

Description

Projective code that is part of a family of \([2^{a+i}+2^i-2^a,3,2^{a+i}-2^a]_{GF(2^a)}\) codes for \(i < a\) constructed using Denniston arcs.

Notes

See corresponding MinT database entry [2].

Cousin

  • Hyperoval code— Denniston codes for \(i=1\) are based on hyperovals in \(PG(2,2^a)\) [2].

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
Projective two-weight codeLinear \(q\)-ary ECC
Parents
Projective two-weight code
Denniston codes are projective two-weight codes on maximal arcs [2][3; Sec. 19.7.3].
Griesmer codeMDS Linear \(q\)-ary OA LRC Distributed-storage \(t\)-design Universally optimal ECC
\(q\)-ary sharp configurationOA Sharp configuration \(t\)-design Universally optimal ECC
The Denniston code is a \(q\)-ary sharp configuration [4; Table 12.1].
Denniston code
Children
Hexacode
A version of the hexacode is recovered for Dennison code parameters \(i=1\) and \(a=2\) [5].

References

[1]
R. H. F. Denniston, “Some maximal arcs in finite projective planes”, Journal of Combinatorial Theory 6, 317 (1969) DOI
[2]
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
[3]
A. E. Brouwer, "Two-weight Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[4]
P. Boyvalenkov, D. Danev, "Linear programming bounds." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[5]
J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016) DOI
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Zoo Code ID: denniston

Cite as:
“Denniston code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/denniston
BibTeX:
@incollection{eczoo_denniston, title={Denniston code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/denniston} }
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Permanent link:
https://errorcorrectionzoo.org/c/denniston

Cite as:

“Denniston code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/denniston

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