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Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory

Abstract : We consider the question of determining the maximum number of Fq-rational points that can lie on a hypersurface of a given degree in a weighted projective space over the finite field Fq, 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 Fq. 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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Submitted on : Thursday, June 22, 2017 - 11:45:55 AM
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Yves Aubry, Wouter Castryck, Sudhir Ghorpade, Gilles Lachaud, Michael O 'Sullivan, et al.. Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory. Algebraic Geometry for Coding Theory and Cryptography, IPAM (UCLA), Feb 2016, Los Angeles, United States. pp.25-61, ⟨10.1007/978-3-319-63931-4_2⟩. ⟨hal-01478729v2⟩

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