Hansen toric code[1,2] 

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.

Parent

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
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Zoo Code ID: toric_classical

Cite as:
“Hansen toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/toric_classical
BibTeX:
@incollection{eczoo_toric_classical,
  title={Hansen toric code},
  booktitle={The Error Correction Zoo},
  year={2022},
  editor={Albert, Victor V. and Faist, Philippe},
  url={https://errorcorrectionzoo.org/c/toric_classical}
}
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Cite as:

“Hansen toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/toric_classical

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/q-ary_digits/ag/varieties/toric_classical.yml.