Klein-quartic code[1]

Description

Evaluation AG code over \(GF(8)\) of rational functions evaluated on points lying in 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 held 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).

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

Zoo code information

Internal code ID: klein_quartic

Your contribution is welcome!

on github.com (edit & pull request)

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Zoo Code ID: klein_quartic

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

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/q-ary_digits/eval/evaluationAG/klein_quartic.yml.