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. [3–8].Notes
See Ref. [8] for various examples and implementations in Magma.Cousin
- Toric code— The toric code is not to be confused with the CSS code constructed from a polynomial evaluation code on a toric variety [9].
Member of code lists
Primary Hierarchy
Parents
Hansen toric codes are polynomial evaluation codes with \(\cal X\) being a toric variety.
Hansen toric code
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. López, C. Rentería-Márquez, and R. H. Villarreal, “Affine cartesian codes”, Designs, Codes and Cryptography 71, 5 (2012) arXiv:1202.0085 DOI
- [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
- [9]
- J. P. Hansen, “Toric Codes, Multiplicative Structure and Decoding”, (2017) arXiv:1702.06569
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