Bounds on the number of rational points of algebraic hypersurfaces  over finite fields, with applications to projective Reed-Muller codes

Daniele Bartoli; Adnen  Sboui; Leo Storme

doi:10.3934/amc.2016010

Article Contents

2016, Volume 10, Issue 2: 355-365. Doi: 10.3934/amc.2016010

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Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes

1.
Dipartimento di Matematica ed Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia
2.
Institut Préparatoire aux Études d'Ingénieurs d'El-Manar, Université Tunis El Manar, Campus universitaire El Manar, B.P.244 El Manar II - 2092 Tunis
3.
Department of Mathematics, Ghent University, Krijgslaan 281 - S22, 9000 Ghent

Received: June 30, 2014

Revised: May 31, 2015

Published: April 2016

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Abstract

We present bounds on the number of points in algebraic curves and algebraic hypersurfaces in $\mathbb{P}^n(\mathbb{F}_q)$ of small degree $d$, depending on the number of linear components contained in such curves and hypersurfaces. The obtained results have applications to the weight distribution of the projective Reed-Muller codes PRM$(q,d,n)$ over the finite field $\mathbb{F}_q$.

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Mathematics Subject Classification: Primary: 51E20, 94B05; Secondary: 05B25.

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References

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