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Designs, Codes and Cryptography

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01.07.2014

The Kakeya problem: a gap in the spectrum and classification of the smallest examples

verfasst von: A. Blokhuis, M. De Boeck, F. Mazzocca, L. Storme

Erschienen in: Designs, Codes and Cryptography | Ausgabe 1/2014

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Abstract

Kakeya sets in the affine plane \(\mathrm AG (2,q)\) are point sets that are the union of lines, one through every point on the line at infinity. The finite field Kakeya problem asks for the size of the smallest Kakeya sets and the classification of these Kakeya sets. In this article we present a new example of a small Kakeya set and we give the classification of the smallest Kakeya sets up to weight \(\frac{q(q+2)}{2}+\frac{q}{4}\), both in case \(q\) even.

Vorheriger Artikel Remarks on polarity designs

Nächster Artikel Maximal partial line spreads of non-singular quadrics

Assmus E.F., Key J.D.: Designs and their codes. Cambridge University Press, New York (1992).

Blokhuis A., Bruen A.A.: The minimal number of lines intersected by a set of \(q+2\) points, blocking sets, and intersecting circles. J. Comb. Theory Ser. A 50, 308–315 (1989).

Blokhuis A., Mazzocca F.: The finite field Kakeya problem. In: Grötschel M., Katona G.O.H. (eds.) Building Bridges Between Mathematics and Computer Science. Bolyai Society Mathematical Studies, Vol. 19, pp. 205–218 (2008).

Dvir Z.: On the size of kakeya sets in finite fields. J. Am. Math. Soc. 22, 1093–1097 (2009).

Dvir Z., Kopparty S., Saraf S., Sudan M.: Extensions to the method of multiplicities, with applications to Kakeya sets and mergers. In: FOCS 09 (to appear) (2009).

Faber X.W.C.: On the finite field Kakeya problem in two dimensions. J. Number Theory 117, 471–481 (2006).

Gács A.: Weiner Zs.: On \((q+t)\)-arcs of type \((0,2, t)\). Des. Codes Cryptogr. 29(1–3), 131–139 (2003).

Hirschfeld J.W.P.: Projective geometries over finite fields. Oxford Mathematical Monographs, 2nd edn. The Clarendon Press, New York, xiv+555 pp (1998).

Hirschfeld J.W.P., Korchmáros G.: Arcs and curves over a finite field. Finite Fields Appl. 5, 393–408 (1999).

10.

Key J.D., McDonough T.P., Mavron V.C.: An upper bound for the minimum weight of the dual codes of desarguesian planes. Eur. J. Comb. 30(1), 220–229 (2009).

11.

Korchmáros G., Mazzocca F.: On \((q+t)\)-arcs of type \((0,2, t)\) in a desarguesian plane of order \(q\). Math. Proc. Camb. Phil. Soc. 108(3), 445–459 (1990).

12.

Lavrauw M., Storme L., Van de Voorde G.: Linear codes from projective spaces. In: Bruen A.A., Wehlau D.L. (eds.) Error-Correcting Codes, Finite Geometries, and Cryptography. AMS Contemporary Mathematics (CONM) Book Series, vol. 523, pp. 185–202 (2010).

13.

Saraf S., Sudan M.: Improved lower bound on the size of Kakeya sets over finite fields. Anal. PDE 1(3), 375–379 (2008)

14.

Segre B.: Ovals in finite projective planes. Can. J. Math. 7, 414–416 (1955).

15.

Tao T.: Poincarés legacies: pages from year two of a mathematical blog. American Mathematical Society, vol. I (2009).

16.

Wolff T.: Recent Work Connected with the Kakeya Problem. Prospects in Mathematics (Princeton, NJ, 1996). American Mathematical Society, Providence, RI, pp. 129–162 (1999).

Titel: The Kakeya problem: a gap in the spectrum and classification of the smallest examples
verfasst von: A. Blokhuis
M. De Boeck
F. Mazzocca
L. Storme
Publikationsdatum: 01.07.2014
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 1/2014
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-012-9790-3

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