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Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory. (English) Zbl 1415.14010
Howe, Everett W. (ed.) et al., Algebraic geometry for coding theory and cryptography, IPAM, Los Angeles, CA, USA, February 2016. Cham: Springer. Assoc. Women Math. Ser. 9, 25-61 (2017).
Summary: We consider the question of determining the maximum number of \(\mathbb{F}_{q}\)-rational points that can lie on a hypersurface of a given degree in a weighted projective space over the finite field \(\mathbb{F}_{q}\), or in other words, the maximum number of zeros that a weighted homogeneous polynomial of a given degree can have in the corresponding weighted projective space over \(\mathbb{F}_{q}\). In the case of classical projective spaces, this question has been answered by J.-P. Serre. In the case of weighted projective spaces, we give some conjectures and partial results. Applications to coding theory are included and an appendix providing a brief compendium of results about weighted projective spaces is also included.
For the entire collection see [Zbl 1387.14013].
MSC:
14G05 Rational points
14G15 Finite ground fields in algebraic geometry
14G50 Applications to coding theory and cryptography of arithmetic geometry
94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
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