Polynomial evaluation code 

Description

Evaluation code of polynomials (or, more generally, rational functions) at points \({\cal P} = \left( P_1,P_2,\cdots,P_n \right)\) on an algebraic variety \(\cal X\) of dimension greater than one (i.e., not an algebraic curve).

Codewords are evaluations of a linear space \(L\) of rational functions \(f\), \begin{align} \left( f(P_1), f(P_2), \cdots, f(P_n) \right)~. \tag*{(1)}\end{align} If the space is taken to be all multivariate polynomials up to some degree, the code is called a Reed-Muller-type code or RM-type code of that order.

Notes

See Refs. [1,2] for reviews.

Parent

  • Evaluation code — Polynomial evaluation codes are evaluation codes for which \(\cal X\) is an algebraic variety of dimension greater than one.

Children

  • Generalized RM (GRM) code — GRM (PRM) codes are multivariate polynomial evaluation codes with \(\cal X\) being the entire \(m\)-dimensional affine (projective) space over \(GF(q)\) [4,5][3; pgs. 44-46].
  • Complete-intersection RM-type code — Complete-intersection RM-type codes are polynomial evaluation codes with \(\cal X\) being a complete intersection.
  • Deligne-Lusztig code — Deligne-Lusztig codes are evaluation AG codes with \(\cal X\) a Deligne-Lusztig curve.
  • Flag-variety code — Flag-variety codes are polynomial evaluation codes with \(\cal X\) being a flag variety.
  • Ruled-surface code — Ruled-surface codes are polynomial evaluation codes with \(\cal X\) being a ruled surface.
  • Segre-variety RM-type code — Segre-variety RM-type codes are polynomial evaluation codes with \(\cal X\) being a Segre variety.
  • Hansen toric code — Hansen toric codes are polynomial evaluation codes with \(\cal X\) being a toric variety.

References

[1]
J. B. Little, “Algebraic geometry codes from higher dimensional varieties”, (2008) arXiv:0802.2349
[2]
D. Jaramillo, M. V. Pinto, and R. H. Villarreal, “Evaluation codes and their basic parameters”, (2020) arXiv:1907.13217
[3]
M. A. Tsfasman and S. G. Vlăduţ, Algebraic-Geometric Codes (Springer Netherlands, 1991) DOI
[4]
S. G. Vléduts and Yu. I. Manin, “Linear codes and modular curves”, Journal of Soviet Mathematics 30, 2611 (1985) DOI
[5]
T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.
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Zoo Code ID: evaluation_polynomial

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

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

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