## 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].

## Parents

## 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 [8][4; Sec. 19.7.3].
- Hill projective-cap code — Hill projective-cap codes are projective two-weight codes on projective caps [4; Table 19.1].

## Cousins

- Ternary Golay code — The dual of the ternary Golay code is a projective two-weight subcode [5,6][4; Ex. 19.3.2][7; Table 7.1].
- Hyperoval code — Codes based on hyperovals in \(PG_{2}(q)\) with even \(q\) are projective two-weight codes [5,6][4; Exam. 19.2.1][7; Table 7.1].

## 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]
- 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]
- 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

## Page edit log

- Victor V. Albert (2023-04-15) — most recent

## Cite as:

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