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
Parent
- 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].
References
- [1]
- J. Justesen et al., “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
Page edit log
- Victor V. Albert (2022-08-05) — most recent
Cite as:
“Plane-curve code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/plane_curve