## 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

## Parent

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

## Children

- Hyperbolic evaluation code
- 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.

## Page edit log

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

## Cite as:

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