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Projective two-weight code

Description

A projective code whose codewords all have one of two possible nonzero Hamming weights.

There is a correspondence between projective two-weight codes, projective two-character sets, and certain strongly regular graphs [2][1; Sec. 19.3.3]. As such, projective two-weight codes have been classified and can be constructed out of quadrics [3,4], maximal arcs and hyperovals, Baer spaces, or Hermitian quadrics [1; Secs. 19.7.1-19.7.5]. There are also several sporadic examples [1; Table 19.1].

Protection

In a projective two-weight code, the difference between the two nonzero weights is a power of the characteristic [1; Sec. 19.3.6].

Cousins

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
Projective two-weight codes are projective codes by definition [1; Sec. 19.2] (see also [8–10]).
Two-weight codeLinear \(q\)-ary ECC
Projective two-weight codes are two-weight codes by definition [1; Def. 19.1] (see also [8–10]).
Projective two-weight code
Children
Ovoid code
The ovoid code is a two-weight projective code [5,6][7; Table 7.1].
Denniston code
Denniston codes are projective two-weight codes on maximal arcs [11][1; Sec. 19.7.3].
Hill projective-cap code
Hill projective-cap codes are projective two-weight codes on projective caps [1; Table 19.1].

References

[1]
A. E. Brouwer, “Two-weight Codes.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[2]
Ph. Delsarte, “Weights of linear codes and strongly regular normed spaces”, Discrete Mathematics 3, 47 (1972) DOI
[3]
A. Ashikhmin and A. Barg, “Binomial moments of the distance distribution: bounds and applications”, IEEE Transactions on Information Theory 45, 438 (1999) DOI
[4]
N. Hamilton, “Strongly regular graphs from differences of quadrics”, Discrete Mathematics 256, 465 (2002) DOI
[5]
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
[6]
A. E. Brouwer and H. Van Maldeghem, Strongly Regular Graphs (Cambridge University Press, 2022) DOI
[7]
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
[8]
A. E. Brouwer and M. van Eupen, Designs, Codes and Cryptography 11, 261 (1997) DOI
[9]
R. Calderbank and W. M. Kantor, “The Geometry of Two-Weight Codes”, Bulletin of the London Mathematical Society 18, 97 (1986) DOI
[10]
D. Jungnickel and V. D. Tonchev, “The classification of antipodal two-weight linear codes”, Finite Fields and Their Applications 50, 372 (2018) DOI
[11]
R. Schürer and W. Ch. Schmid. “Denniston Codes.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. mint.sbg.ac.at/desc_CDenniston.html
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Zoo Code ID: projective_two_weight

Cite as:
“Projective two-weight code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/projective_two_weight
BibTeX:
@incollection{eczoo_projective_two_weight, title={Projective two-weight code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/projective_two_weight} }
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Permanent link:
https://errorcorrectionzoo.org/c/projective_two_weight

Cite as:

“Projective two-weight code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/projective_two_weight

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