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

Daniele Bartoli (UGent) , Adnen Sboui and Leo Storme (UGent)
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Abstract
We present bounds on the number of points in algebraic curves and algebraic hypersurfaces in P-n(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 IFq.
Keywords
quadrics, Algebraic varieties, small weight codewords, intersections, projective Reed-Muller codes, P-N(F-Q), ARCS

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Chicago
Bartoli, Daniele, Adnen Sboui, and Leo Storme. 2016. “Bounds on the Number of Rational Points of Algebraic Hypersurfaces over Finite Fields, with Applications to Projective Reed-Muller Codes.” Advances in Mathematics of Communications 10 (2): 355–365.
APA
Bartoli, D., Sboui, A., & Storme, L. (2016). Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes. ADVANCES IN MATHEMATICS OF COMMUNICATIONS, 10(2), 355–365.
Vancouver
1.
Bartoli D, Sboui A, Storme L. Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes. ADVANCES IN MATHEMATICS OF COMMUNICATIONS. 2016;10(2):355–65.
MLA
Bartoli, Daniele, Adnen Sboui, and Leo Storme. “Bounds on the Number of Rational Points of Algebraic Hypersurfaces over Finite Fields, with Applications to Projective Reed-Muller Codes.” ADVANCES IN MATHEMATICS OF COMMUNICATIONS 10.2 (2016): 355–365. Print.
@article{8028382,
  abstract     = {We present bounds on the number of points in algebraic curves and algebraic hypersurfaces in P-n(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 IFq.},
  author       = {Bartoli, Daniele and Sboui, Adnen and Storme, Leo},
  issn         = {1930-5346},
  journal      = {ADVANCES IN MATHEMATICS OF COMMUNICATIONS},
  keywords     = {quadrics,Algebraic varieties,small weight codewords,intersections,projective Reed-Muller codes,P-N(F-Q),ARCS},
  language     = {eng},
  number       = {2},
  pages        = {355--365},
  title        = {Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes},
  url          = {http://dx.doi.org/10.3934/amc.2016010},
  volume       = {10},
  year         = {2016},
}

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