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Plane-curve evaluation code[1]

Description

Evaluation AG code of bivariate polynomials of some finite maximum degree, evaluated at points lying on an affine or projective plane curve.

Protection

Bezout’s theorem yields parameters \([n,k,d]\), which depend on the polynomial used to define the plane curve as well as the maximum degree of the polynomials used for evaluation [2; pg. 883]. Distance bounds can be derived from how the plane curve is embedded in the ambient projective space [3; Thm. 4.1].

Decoding

Generalization of the Peterson algorithm for BCH codes [1,4,5].

Cousin

Member of code lists

Primary Hierarchy

Classical Domain
Galois-field Kingdom
\(q\)-ary codeECC
Additive \(q\)-ary codeECC
Linear \(q\)-ary codeECC
Evaluation codeAG ECC
Evaluation AG code
Parents
Evaluation AG code
Plane-curve evaluation codes are evaluation AG codes of bivariate polynomials with \(\cal X\) being an affine plane curve [6][2; Thm. 2.27].
Plane-curve evaluation code

References

[1]
J. Justesen, K. J. Larsen, H. E. Jensen, A. Havemose, and T. Hoholdt, “Construction and decoding of a class of algebraic geometry codes”, IEEE Transactions on Information Theory 35, 811 (1989) DOI
[2]
T. Høholdt, J. H. van Lint, and R. Pellikaan, “Algebraic geometry codes”, in Handbook of Coding Theory, Vol. I, Part 1, eds. V. S. Pless and W. C. Huffman (Elsevier, 1998), pp. 871-961
[3]
A. Couvreur, “The dual minimum distance of arbitrary-dimensional algebraic–geometric codes”, Journal of Algebra 350, 84 (2012) arXiv:0905.2345 DOI
[4]
A. N. Skorobogatov and S. G. Vladut, “On the decoding of algebraic-geometric codes”, IEEE Transactions on Information Theory 36, 1051 (1990) DOI
[5]
V. Yu. Krachkovsky, (1988) “Decoding of codes on algebraic curves”, Proceedings of the IX All-Union Conference on the Theory of Coding and Information Transmission, Moscow-Odessa: USSR Academy of Sciences, 1988 Part 2, pp. 143–146
[6]
M. A. Tsfasman and S. G. Vlăduţ, Algebraic-Geometric Codes (Springer Netherlands, 1991) DOI
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Zoo Code ID: plane_curve

Cite as:
“Plane-curve evaluation code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/plane_curve, arXiv:2606.11484
BibTeX:
@incollection{eczoo_plane_curve,
title={Plane-curve evaluation code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/plane_curve}
}
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Permanent link:
https://errorcorrectionzoo.org/c/plane_curve

Cite as:

“Plane-curve evaluation code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/plane_curve, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/ag/evaluationAG/plane_curve.yml.