Klein-quartic code[1]
Description
Evaluation AG code over \(GF(8)\) of rational functions evaluated on points lying on the Klein quartic, which is defined by the equation \(x^3 y + y^3 z + z^3 x = 0\) ([2], Ex. 2.75).
Protection
Dimension \(k=8\) and distance \(d \geq 13\). Concatenation with the \([4,3,2]\) single parity check code, conversion to a binary code by expressing \(GF(8)\) elements as vectors over \(GF(2)\), and puncturing yields a \([91,24,25]\) binary code that set the world record for codes of length 91 [3].
Parent
- Evaluation AG code — Klein-quartic codes are evaluation AG codes with \(\cal X\) being the Klein quartic ([2], Ex. 2.75)[4].
Cousin
- Group-algebra code — Some Klein-quartic codes are group-algebra codes [5; Remark 16.4.14].
References
- [1]
- J. Hansen, “Codes on the Klein quartic, ideals, and decoding (Corresp.)”, IEEE Transactions on Information Theory 33, 923 (1987) DOI
- [2]
- T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.
- [3]
- A. M. Barg, G. L. Katsman, M. A. Tsfasman, “Algebraic-Geometric Codes from Curves of Small Genus”, Probl. Peredachi Inf., 23:1 (1987), 42–46; Problems Inform. Transmission, 23:1 (1987), 34–38
- [4]
- M. A. Tsfasman and S. G. Vlăduţ, Algebraic-Geometric Codes (Springer Netherlands, 1991) DOI
- [5]
- W. Willems, "Codes in Group Algebras." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
Page edit log
- Victor V. Albert (2022-08-03) — most recent
Cite as:
“Klein-quartic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/klein_quartic