Showing papers by "Gábor Korchmáros published in 2009"
TL;DR: In this paper, it was shown that the Deligne-Lusztig curves associated to the algebraic groups of type \({A_2,\,^2B_2} and \({^2G_2}) defined over finite fields all have the maximum number of rational points allowed by the Weil "explicit formulas".
Abstract: It has been known for a long time that the Deligne–Lusztig curves associated to the algebraic groups of type \({^2A_2,\,^2B_2}\) and \({^2G_2}\) defined over the finite field \({\mathbb {F}_n}\) all have the maximum number of \({\mathbb {F}_n}\)-rational points allowed by the Weil “explicit formulas”, and that these curves are \({\mathbb {F}_{q^2}}\)-maximal curves over infinitely many algebraic extensions \({\mathbb {F}_{q^2}}\) of \({\mathbb {F}_n}\). Serre showed that an \({\mathbb {F}_{q^2}}\)-rational curve which is \({\mathbb {F}_{q^2}}\)-covered by an \({\mathbb {F}_{q^2}}\)-maximal curve is also \({\mathbb {F}_{q^2}}\)-maximal. This has posed the problem of the existence of \({\mathbb {F}_{q^2}}\)-maximal curves other than the Deligne–Lusztig curves and their \({\mathbb {F}_{q^2}}\)-subcovers, see for instance Garcia (On curves with many rational points over finite fields. In: Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, pp. 152–163. Springer, Berlin, 2002) and Garcia and Stichtenoth (A maximal curve which is not a Galois subcover of the Hermitan curve. Bull. Braz. Math. Soc. (N.S.) 37, 139–152, 2006). In this paper, a positive answer to this problem is obtained. For every q = n3 with n = pr > 2, p ≥ 2 prime, we give a simple, explicit construction of an \({\mathbb {F}_{q^2}}\)-maximal curve \({\mathcal {X}}\) that is not \({\mathbb {F}_{q^2}}\)-covered by any \({\mathbb {F}_{q^2}}\)-maximal Deligne–Lusztig curve. Furthermore, the \({\mathbb {F}_{q^2}}\)-automorphism group Aut\({(\mathcal {X})}\) has size n3(n3 + 1)(n2 − 1)(n2 − n + 1). Interestingly, \({\mathcal {X}}\) has a very large \({\mathbb {F}_{q^2}}\)-automorphism group with respect to its genus \({g = \frac{1}{2}\,(n^3 + 1)(n^2 - 2) + 1}\).
134 citations
TL;DR: In PG(2,q^2) with q>3, the intersection pattern of U and @W is determined where U is a Hermitian unital and@W is a classical oval.
15 citations
TL;DR: Theorem 6.1 as discussed by the authors shows that there exists a (1 + )-arc in PG(2, q) with r odd and q ≥ 3 (mod 4) points with a conic.
Abstract: The sporadic complete 12-arc in PG(2, 13) contains eight points from a conic. In PG(2,q) with q>13 odd, all known complete k-arcs sharing exactly ½(q+3) points with a conic have size at most ½(q+3)+2, with only two exceptions, both due to Pellegrino, which are complete (½(q+3)+3) arcs, one in PG(2, 19) and another in PG(2, 43). Here, three further exceptions are exhibited, namely a complete (½(q+3)+4)-arc in PG(2, 17), and two complete (½(q+3)+3)-arcs, one in PG(2, 27) and another in PG(2, 59). The main result is Theorem 6.1 which shows the existence of a (½(qr+3)+3)-arc in PG(2,qr) with r odd and q≡3 (mod 4) sharing ½(qr+3) points with a conic, whenever PG(2,q) has a (½(qr+3)+3)-arc sharing ½(qr+3) points with a conic. A survey of results for smaller q obtained with the use of the MAGMA package is also presented. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 25–47, 2010
9 citations
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TL;DR: In this article, it was shown that if all lines from two classes of a 3-net are tangent to the same irreducible conic, then all lines of the third class are concurrent.
Abstract: We investigate finite 3-nets embedded in a projective plane over a (finite or infinite) field of any characteristic p. Such an embedding is regular when each of the three classes of the 3-net comprises concurrent lines, and irregular otherwise. It is completely irregular when no class of the 3-net consists of concurrent lines. We are interested in embeddings of 3-nets which are irregular but the lines of one class are concurrent. For an irregular embedding of a 3-net of order n greater than 4 we prove that, if all lines from two classes are tangent to the same irreducible conic, then all lines from the third class are concurrent. We also prove the converse provided that the order n of the 3-net is smaller than p. In the complex plane, apart from a sporadic example of order n=5 due to Stipins, each known irregularly embedded 3-net has the property that all its lines are tangent to a plane cubic curve. Actually, the procedure of constructing irregular 3-nets with this property works over any field. In positive characteristic, we present some more examples for n greater than 4 and give a complete classification for n=4.