Description
Evaluation code of a linear space of polynomials evaluated on points lying on an affine or projective toric variety. If the space is taken to be all polynomials up to some degree, the code is called a toric RM-type code of that degree.
Protection
Parameters of toric RM-type codes and various generalizations have been determined in Refs. [3][4][5][6][7][8].
Notes
See Ref. [8] for various examples and implementations in Magma.
Parent
- Polynomial evaluation code — Hansen toric codes are polynomial evaluation codes with \(\cal X\) being a toric variety.
References
- [1]
- J. P. Hansen, “Toric Surfaces and Error-correcting Codes”, Coding Theory, Cryptography and Related Areas 132 (2000) DOI
- [2]
- D. Joyner, “Toric codes over finite fields”, (2003) arXiv:math/0208155
- [3]
- D. Ruano, “On the Parameters of r-dimensional Toric Codes”, (2005) arXiv:math/0512285
- [4]
- J. Little and H. Schenck, “Toric surface codes and Minkowski sums”, (2006) arXiv:math/0507598
- [5]
- E. Sarmiento, M. V. Pinto, and R. H. Villarreal, “The minimum distance of parameterized codes on projective tori”, (2011) arXiv:1009.4966
- [6]
- H. H. Lopez, C. Renteria, and R. H. Villarreal, “Affine cartesian codes”, (2012) arXiv:1202.0085
- [7]
- P. Beelen and M. Datta, “Generalized Hamming weights of affine cartesian codes”, (2017) arXiv:1706.02114
- [8]
- D. Jaramillo, M. V. Pinto, and R. H. Villarreal, “Evaluation codes and their basic parameters”, (2020) arXiv:1907.13217
Page edit log
- Victor V. Albert (2022-08-11) — most recent
Cite as:
“Hansen toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/toric_classical