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Plane-curve 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].

Member of code lists

Primary Hierarchy

Classical Domain
Galois-field Kingdom
\(q\)-ary codeECC
Additive \(q\)-ary codeECC
Linear \(q\)-ary codeECC
Evaluation code
Evaluation AG codeAG ECC
Parents
Evaluation AG code
Plane-curve codes are evaluation AG codes of bivariate polynomials with \(\cal X\) being an affine plane curve ([2], Thm. 2.27)[6].
Plane-curve 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, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), 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. Krachkovskii, "Decoding of codes on algebraic curves," (in Russian), Conference Odessa, 1988.
[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 code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/plane_curve
BibTeX:
@incollection{eczoo_plane_curve, title={Plane-curve code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/plane_curve} }
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Permanent link:
https://errorcorrectionzoo.org/c/plane_curve

Cite as:

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

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