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Quadric code[1,2]

Description

Evaluation code of polynomials evaluated on points lying on a quadric hypersurface.

Cousin

  • Quadric tower code— The classical code underlying a quadric tower code is the evaluation code of polynomials of degree at most \(j-2\) on the points of a quadric hypersurface, the affine binary case of the RM codes associated to algebraic varieties [2]. The \(j=3\) member is the three-weight \([28,7,12]\) code, whose nonzero weights \(12\), \(16\), and \(28\) place it among the two- and three-weight codes associated to hyperquadrics [1]; higher members use polynomials of degree above one and are no longer few-weight.

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
Polynomial evaluation code
Flag-variety code
Parents
Flag-variety code
Quadric codes are flag-variety evaluation codes with the flag variety being a quadric hypersurface.
Quadric code

References

[1]
J. Wolfmann, “Codes projectifs a deux ou trois poids associfs aux hyperquadriques d’une geometrie finie”, Discrete Mathematics 13, 185 (1975) DOI
[2]
Y. Aubry, “Reed-Muller codes associated to projective algebraic varieties”, Lecture Notes in Mathematics 4 (1992) DOI
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Zoo Code ID: quadric

Cite as:
“Quadric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/quadric, arXiv:2606.11484
BibTeX:
@incollection{eczoo_quadric,
title={Quadric code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/quadric}
}
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Permanent link:
https://errorcorrectionzoo.org/c/quadric

Cite as:

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

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