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In this paper, we show that there are at least cq disjoint blocking sets in PG(2,q), where c ≈ 1/3. The result also extends to some non‐Desarguesian planes of order q. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 149–158, 2006

The size of large minimal blocking sets is bounded by the Bruen–Thas upper bound. The bound is sharp when q is a square. Here the bound is improved if q is a non‐square. On the other hand, we present some constructions of reasonably large minimal blocking sets in planes of non‐prime order. The construction can be regarded as a generalization of Buekenhout's construction of unitals. For example, if q is a cube, then our construction gives minimal blocking sets of size q4/3 + 1 or q4/3 + 2. Density results for the spectrum of minimal blocking sets in Galois planes of non‐prime order is also presented. The most attractive case is when q is a square, where we show that there is a minimal blocking set for any size from the interval $[4q\,{\rm log}\, q, q\sqrt q-q+2\sqrt q]$. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 25–41, 2005.

In this paper we construct maximal partial spreads in PG(3, q) which are a log qfactor larger than the best known lower bound. For n≥ 5 we also construct maximal partial spreads in PG(n, q) of each size between cnqn− 2log qand c′ qn− 1.

We show that the cardinality of a nonempty set of points without tangents in the desarguesian projective plane PG(2, q), qeven, is at least q+ 1 + $$ \sqrt {q/6} $$ provided that the set is not of even type.

The author is a graduated student of Pázmány Péter Catholic University who carried out research in Rome for three months in 1999. In this period he found in the General Archives of the Dominican Order several documents related to Hungary which arrived to Rome presumably after the second world war. In this paper he is presenting two of these documents that he is now studying on photographs. The two Latin texts are from the beginning of the XVIthcentury and contain valuable informations for researchers of the history of asile right.

We show that small blocking sets in PG(n, q) with respect to hyperplanes intersect every hyperplane in 1 modulo ppoints, where q=ph. The result is then extended to blocking sets with respect to k-dimensional subspaces and, at least when p>2, to intersections with arbitrary subspaces not just hyperplanes. This can also be used to characterize certain non-degenerate blocking sets in higher dimensions. Furthermore we determine the possible sizes of small minimal blocking sets with respect to k-dimensional subspaces.

Let GF(q) be the Galois field of orderq=ph, and letm≥3 be an integer. An explicit formula for the number of GF(qm)-rational points of the Fermat curveXn+Yn+Zn=0 is given whenndivides (qm−1)/(q−1) andpis sufficiently large with respect to (qm−1)/(n(q−1)).

Abstract: In this paper we give k-sets (k>-q+1) of the Hall planes with q-radicq nuclei. This shows that the results of BICHARA — KORCHMÁROS [1] and WETTL [16] cannot be extended to non-desarguesian planes. We construct some inherited arcs in the Hall planes as well. For example complete (q–1)-arcs of the Hall planes of odd order will be constructed.

In this paper the number of directions determined by a set ofq−npoints ofAG(2, q) is studied. To such a set we associate a curve of degreenand show that its linear components correspond to points that can be added to the set without changing the set of determined directions. The existence of linear components is guaranteed by Weil's theorem concerning the number of GF(q)-rational points of an absolutely irreducible curve, ifnis small enough.

We study minimal blocking sets inPG(2, q) havingq+mpoints outside some fixed line. If0

In this paper we show that blocking sets of cardinality less than 3(q+ 1)/2 (q=pn) in Desarguesian projective planes intersect every line in 1 moduloppoints. It is also shown that the cardinality of a blocking set must lie in a few relatively short intervals. This is similar to previous results of Rédei, which were proved for a special class of blocking sets. In the particular caseq=p2, the above result implies that a nontrivial blocking set either contains a Baer-subplane or has size at least 3(q+ 1)/2; and this result is sharp. As a by-product, new proofs are given for the Jamison, Brouwer-Schrijver theorem on blocking sets in Desarguesian affine planes, and for Blokhuis' theorem on blocking sets in Desarguesian projective planes.

Let H be a subgroup of a cyclic Singer group of PG (n,q). In this paper we study the following problem: When is a point orbit of H a cap? A necessary and sufficient condition for this is derived and used to give a short proof of some results by Ebert [6] and to show that small orbits are typically caps.

In this paper we construct a class of k-arcs in PG(2, q), q=ph, h>1, p?3 and prove its completeness for h large enough. The main result states that this class contains complete k-arcs with $$k \leqslant 2 \cdot q^{{9 \mathord{\left/ {\vphantom {9 {10}}} \right. \kern-\nulldelimiterspace} {10}}} {\text{ }}\left( {10{\text{ divides }}h{\text{ and }}q{\text{ }} \geqslant {\text{ }}q_{\text{0}} } \right).$$ Such complete k-arcs are the unique known complete k-arcs with $$k \leqslant {q \mathord{\left/ {\vphantom {q 4}} \right. \kern-\nulldelimiterspace} 4}.$$

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Abstract: The theorem of B. Segre mentioned in the title states that a complete arc of PG(2,q),q even which is not a hyperoval consists of at mostq−√q+1 points. In the first part of our paper we prove this theorem to be sharp forq=s 2 by constructing completeq−√q+1-arcs. Our construction is based on the cyclic partition of PG(2,q) into disjoint Baer-subplanes. (See Bruck [1]). In his paper [5] Kestenband constructed a class of (q−√q+1)-arcs but he did not prove their completeness. In the second part of our paper we discuss the connections between Kestenband’s and our constructions. We prove that these constructions result in isomorphic (q−√q+1)-arcs. The proof of this isomorphism is based on the existence of a traceorthogonal normal basis in GF(q 3) over GF(q), and on a representation of GF(q)3 in GF(q 3)3 indicated in Jamison [4].

A subsetSof a finite projective plane of orderqis called a blocking set ifSmeets every line but contains no line. For the size of an inclusion-minimal blocking setq+ $$\sqrt q $$ +≤∣S∣≤q $$\sqrt q $$ +1 holds ([6]). Ifqis a square, then inPG(2,q) there are minimal blocking sets with cardinalityq $$\sqrt q $$ +1. Ifqis not a square, then the various constructions known to the author yield minimal blocking sets with less than 3qpoints. In the present note we show that inPG(2,q),q≡1 (mod 4) there are minimal blocking sets having more thanqlog2q/2 points. The blocking sets constructed in this note contain the union ofkconics, wherek≤log2q/2. A slight modification of the construction works forq≡3 (mod 4) and gives the existence of minimal blocking sets of sizecqlog2qfor some constantc. As a by-product we construct minimal blocking sets of cardinalityq $$\sqrt q $$ +1, i.e. unitals, in Galois planes of square order. Since these unitals can be obtained as the union of $$\sqrt q $$ parabolas, they are not classical.