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.

Member of code lists

Primary Hierarchy

Classical Domain

Galois-field Kingdom

\(q\)-ary codeECC

Additive \(q\)-ary codeECC

Linear \(q\)-ary codeECC

Evaluation code

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

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

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

Polynomial evaluation code

Description

Notes

Member of code lists

Primary Hierarchy

References

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