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

Daniele Bartoli | Zendy; Adnen Sboui | Zendy; Leo Storme | Zendy

Open Access

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

Author(s) -

Daniele Bartoli,

Adnen Sboui,

Leo Storme

Publication year - 2016

Publication title -

advances in mathematics of communications

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.601

H-Index - 26

eISSN - 1930-5346

pISSN - 1930-5338

DOI - 10.3934/amc.2016010

Subject(s) - mathematics , finite field , algebraic number , algebraic curve , projective test , degree (music) , field (mathematics) , combinatorics , discrete mathematics , algebraic geometry , algebraic number field , pure mathematics , mathematical analysis , physics , acoustics

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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