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1.
A unital U with parameter q is a 2 – (q 3 + 1, q + 1, 1) design. If a point set U in PG(2, q 2) together with its (q + 1)-secants forms a unital, then U is called a Hermitian arc. Through each point p of a Hermitian arc H there is exactly one line L having with H only the point p in common; this line L is called the tangent of H at p. For any prime power q, the absolute points and nonabsolute lines of a unitary polarity of PG(2, q 2) form a unital that is called the classical unital. The points of a classical unital are the points of a Hermitian curve in PG(2, q 2).Let H be a Hermitian arc in the projective plane PG(2, q 2). If tangents of H at collinear points of H are concurrent, then H is a Hermitian curve. This result proves a well known conjecture on Hermitian arcs.  相似文献   

2.
We show that a suitable 2-dimensional linear system of Hermitian curves of PG(2,q 2) defines a model for the Desarguesian plane PG(2,q). Using this model we give the following group-theoretic characterization of the classical unitals. A unital in PG(2,q 2) is classical if and only if it is fixed by a linear collineation group of order 6(q + 1)2 that fixes no point or line in PG(2,q 2).  相似文献   

3.
An infinite family of complete (q 2 + q + 8)/2-caps is constructed in PG(3, q) where q is an odd prime ≡ 2 (mod 3), q ≥ 11. This yields a new lower bound on the second largest size of complete caps. A variant of our construction also produces one of the two previously known complete 20-caps in PG(3, 5). The associated code weight distribution and other combinatorial properties of the new (q 2 + q + 8)/2-caps and the 20-cap in PG(3, 5) are investigated. The updated table of the known sizes of the complete caps in PG(3, q) is given. As a byproduct, we have found that the unique complete 14-arc in PG(2, 17) contains 10 points on a conic. Actually, this shows that an earlier general result dating back to the Seventies fails for q = 17.   相似文献   

4.
Some geometry of Hermitian matrices of order three over GF(q2) is studied. The variety coming from rank 2 matrices is a cubic hypersurface M73of PG(8,q ) whose singular points form a variety H corresponding to all rank 1 Hermitian matrices. BesideM73 turns out to be the secant variety of H. We also define the Hermitian embedding of the point-set of PG(2, q2) whose image is exactly the variety H. It is a cap and it is proved that PGL(3, q2) is a subgroup of all linear automorphisms of H. Further, the Hermitian lifting of a collineation of PG(2, q2) is defined. By looking at the point orbits of such lifting of a Singer cycle of PG(2, q2) new mixed partitions of PG(8,q ) into caps and linear subspaces are given.  相似文献   

5.
The unitals in the Hall plane are studied by deriving PG(2,q 2)and observing the effect on the unitals of PG(2,q 2).The number of Buekenhout and Buekenhout-Metz unitals in the Hall plane is determined. As a corollary we show that the classical unital is not embeddble in the Hall plane as a Buekenhout unital and that the Buekenhout unitals of H(q 2)are not embeddable as Buekenhout unitals in the Desarguesian plane. Finally, we generalize this technique to other translation planes.  相似文献   

6.
We first note that each element of a symplectic spread of PG(2n − 1, 2 r ) either intersects a suitable nonsingular quadric in a subspace of dimension n − 2 or is contained in it, then we prove that this property characterises symplectic spreads of PG(2n − 1, 2 r ). As an application, we show that a translation plane of order q n , q even, with kernel containing GF(q), is defined by a symplectic spread if and only if it contains a maximal arc of the type constructed by Thas (Europ J Combin 1:189–192, 1980).  相似文献   

7.
In this paper, we give characterizations of the classical generalized quadrangles H(3, q 2) and H(4, q 2), embedded in PG(3, q 2) and PG(4, q 2), respectively. The intersection numbers with lines and planes characterize H(3, q 2), and H(4, q 2) is characterized by its intersection numbers with planes and solids. This result is then extended to characterize all Hermitian varieties in dimension at least 4 by their intersection numbers with planes and solids.   相似文献   

8.
In this paper we show that starting from a symplectic semifield spread S{\mathcal{S}} of PG(5, q), q odd, another symplectic semifield spread of PG(5, q) can be obtained, called the symplectic dual of S{\mathcal{S}}, and we prove that the symplectic dual of a Desarguesian spread of PG(5, q) is the symplectic semifield spread arising from a generalized twisted field. Also, we construct a new symplectic semifield spread of PG(5, q) (q = s 2, s odd), we describe the associated commutative semifield and deal with the isotopy issue for this example. Finally, we determine the nuclei of the commutative pre-semifields constructed by Zha et al. (Finite Fields Appl 15(2):125–133, 2009).  相似文献   

9.
Let Ω and be a subset of Σ = PG(2n−1,q) and a subset of PG(2n,q) respectively, with Σ ⊂ PG(2n,q) and . Denote by K the cone of vertex Ω and base and consider the point set B defined by
in the André, Bruck-Bose representation of PG(2,qn) in PG(2n,q) associated to a regular spread of PG(2n−1,q). We are interested in finding conditions on and Ω in order to force the set B to be a minimal blocking set in PG(2,qn) . Our interest is motivated by the following observation. Assume a Property α of the pair (Ω, ) forces B to turn out a minimal blocking set. Then one can try to find new classes of minimal blocking sets working with the list of all known pairs (Ω, ) with Property α. With this in mind, we deal with the problem in the case Ω is a subspace of PG(2n−1,q) and a blocking set in a subspace of PG(2n,q); both in a mutually suitable position. We achieve, in this way, new classes and new sizes of minimal blocking sets in PG(2,qn), generalizing the main constructions of [14]. For example, for q = 3h, we get large blocking sets of size qn + 2 + 1 (n≥ 5) and of size greater than qn+2 + qn−6 (n≥ 6). As an application, a characterization of Buekenhout-Metz unitals in PG(2,q2k) is also given.  相似文献   

10.
In this paper, we study the p-ary linear code C(PG(n,q)), q = p h , p prime, h ≥ 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n,q), and its dual code. We link the codewords of small weight of this code to blocking sets with respect to lines in PG(n,q) and we exclude all possible codewords arising from small linear blocking sets. We also look at the dual code of C(PG(n,q)) and we prove that finding the minimum weight of the dual code can be reduced to finding the minimum weight of the dual code of points and lines in PG(2,q). We present an improved upper bound on this minimum weight and we show that we can drop the divisibility condition on the weight of the codewords in Sachar’s lower bound (Geom Dedicata 8:407–415, 1979). G. Van de Voorde’s research was supported by the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen).  相似文献   

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