Polynomial evaluation code

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.

Cousin

Multiplicity code— A multiplicity code is a generalization of an $m$-variate polynomial evaluation code based on evaluating polynomials together with their Hasse derivatives up to order $s-1$ at all points in $\mathbb{F}_q^m$.

Member of code lists

Primary Hierarchy

Classical Domain

Galois-field Kingdom

$q$-ary codeECC

Additive $q$-ary codeECC

Linear $q$-ary codeECC

Evaluation codeAG ECC

Parents

Evaluation code

Polynomial evaluation codes are evaluation codes for which $\cal X$ is an algebraic variety of dimension greater than one.

Polynomial evaluation code

Children

Hyperbolic evaluation code

A hyperbolic evaluation code is an evaluation code over polynomials in two variables.

Generalized RM (GRM) code

GRM (PRM) codes are multivariate polynomial evaluation codes with $\cal X$ being the entire $m$-dimensional affine (projective) space over $\mathbb{F}_q$ [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 variety.

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, “Algebraic geometry codes”, in Handbook of Coding Theory, Vol. I, Part 1, eds. V. S. Pless and W. C. Huffman (Elsevier, 1998), pp. 871-961

Page edit log

Victor V. Albert (2026-06-08) — most recent
Victor V. Albert (2022-08-11)

Cite as:

“Polynomial evaluation code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/evaluation_polynomial, arXiv:2606.11484

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

Polynomial evaluation code

Description

Notes

Cousin

Member of code lists

Primary Hierarchy

References

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