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2017 | OriginalPaper | Chapter

2. Hypersurfaces in Weighted Projective Spaces Over Finite Fields with Applications to Coding Theory

Authors : Yves Aubry, Wouter Castryck, Sudhir R. Ghorpade, Gilles Lachaud, Michael E. O’Sullivan, Samrith Ram

Published in: Algebraic Geometry for Coding Theory and Cryptography

Publisher: Springer International Publishing

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Abstract

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.

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Title: Hypersurfaces in Weighted Projective Spaces Over Finite Fields with Applications to Coding Theory
Authors: Yves Aubry
Wouter Castryck
Sudhir R. Ghorpade
Gilles Lachaud
Michael E. O’Sullivan
Samrith Ram
Publisher: Springer International Publishing
Book: Algebraic Geometry for Coding Theory and Cryptography
Print ISBN: 978-3-319-63930-7

Electronic ISBN: 978-3-319-63931-4

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-3-319-63931-4_2

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