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 [1]. As such, projective two-weight codes have been classified and can be constructed out of quadrics [2,3], maximal arcs and hyperovals, Baer spaces, or Hermitian quadrics [4]. There are also several sporadic examples [4; Table 19.1].

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References

[1]
Ph. Delsarte, “Weights of linear codes and strongly regular normed spaces”, Discrete Mathematics 3, 47 (1972) DOI
[2]
A. Ashikhmin and A. Barg, “Binomial moments of the distance distribution: bounds and applications”, IEEE Transactions on Information Theory 45, 438 (1999) DOI
[3]
N. Hamilton, “Strongly regular graphs from differences of quadrics”, Discrete Mathematics 256, 465 (2002) DOI
[4]
A. E. Brouwer, "Two-weight Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[5]
Rudolf Schürer and Wolfgang Ch. Schmid. “Denniston Codes.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://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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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.yml.