The automorphism group of a complete (q − 1)-arc in PG(2,q)

The automorphism group of the set of 12 points associated with an apolar system of conics is determined. A complete (q ? 1)-arc for q = 13 can be obtained as a special case. The orbits of its automorphism group are also described. © 1994 John Wiley & Sons, Inc.     

The André/Bruck and Bose representation of conics in Baer subplanes of PG(2,q 2)

The André/Bruck and Bose representation ([1], [5,6]) of PG(2,q ²) in PG(4,q) is a tool used by many authors in the proof of recent results. In this paper the André/Bruck and Bose representation of conics in Baer subplanes of PG(2,q ²) is determined. It is proved that a non-degenerate conic in a Baer subplane of PG(2,q ²) is a normal rational curve of order 2, 3, or 4 in the André/Bruck and Bose representation. Moreover the three possibilities (classes) are examined and we classify the conics in each class. Received 1 September 1999; revised 17 July 2000.     

Twisted cubic and point-line incidence matrix in $$\mathrm {PG}(3,q)$$ PG ( 3 , q )

Designs, Codes and Cryptography - We consider the structure of the point-line incidence matrix of the projective space $$\mathrm {PG}(3,q)$$ connected with orbits of points and lines under the...     

Unitals of PG(2,q2) Containing Conics

A unital in PG(2, q²) is a set of points such that each line meets in 1 or points. The well‐known example is the classical unital consisting of all absolute points of a unitary polarity of PG(2, q²). Unitals other than the classical one also exist in PG(2, q²) for every . Actually, all known unitals are of Buekenhout–Metz type [see F. Buekenhout, Geom Dedicata 5 (1976), 189–194, R. Metz, Geom Dedicata 8 (1979), 125–126.], and they can be obtained by a construction due to F. Buekenhout, (Geom Dedicata 5 (1976), 189–194).. The unitals constructed by R. D. Baker and G. L. Ebert (J Combin Theory Ser A 60 (1992), 67–84), and independently by J. W. P. Hirschfeld and T. Sz?nyi (Discrete Math 97 (1991), 229–242), are the union of q conics. Our Theorem  1.1 shows that this geometric property characterizes the Baker–Ebert–Hirschfeld–Sz?nyi unitals. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 101–111, 2013     

On sizes of complete caps in projective spaces PG(n, q) and arcs in planes PG(2, q)

More than thirty new upper bounds on the smallest size t ₂(2, q) of a complete arc in the plane PG(2, q) are obtained for (169 ≤ q ≤ 839. New upper bounds on the smallest size t ₂(n, q) of the complete cap in the space PG(n, q) are given for n = 3 and 25 ≤ q ≤ 97, q odd; n = 4 and q = 7, 8, 11, 13, 17; n = 5 and q = 5, 7, 8, 9; n = 6 and q = 4, 8. The bounds are obtained by computer search for new small complete arcs and caps. New upper bounds on the largest size m ₂(n, q) of a complete cap in PG(n, q) are given for q = 4, n = 5, 6, and q = 3, n = 7, 8, 9. The new lower bound 534 ≤ m ₂(8, 3) is obtained by finding a complete 534-cap in PG(8, 3). Many new sizes of complete arcs and caps are obtained. The updated tables of upper bounds for t ₂(n, q), n ≥ 2, and of the spectrum of known sizes for complete caps are given. Interesting complete caps in PG(3, q) of large size are described. A proof of the construction of complete caps in PG(3, 2 ^h ) announced in previous papers is given; this is modified from a construction of Segre. In PG(2, q), for q = 17, δ = 4, and q = 19, 27, δ = 3, we give complete ${(\frac{1}{2}(q + 3) + \delta)}$ -arcs other than conics that share ${\frac{1}{2}(q + 3)}$ points with an irreducible conic. It is shown that they are unique up to collineation. In PG(2, q), ${{q \equiv 2}}$ (mod 3) odd, we propose new constructions of ${\frac{1}{2} (q + 7)}$ -arcs and show that they are complete for q ≤ 3701.     

Blocking sets in PG(2, q n ) from cones of PG(2n, q)

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

On symplectic semifield spreads of $$\hbox {PG}(5, q^2)$$PG(5, q2), q even

We prove that the only symplectic semifield spreads of \(\hbox {PG}(5,q^2)\), \(q\ge 2^{14}\) even, whose associated semifield has center containing \({\mathbb F}_q\), is the Desarguesian spread, by proving that the only \({\mathbb F}_q\)-linear set of rank 6 disjoint from the secant variety of the Veronese surface of \(\hbox {PG}(5,q^2)\) is a plane with three points of the Veronese surface of \(\hbox {PG}(5,q^6){\setminus } \hbox {PG}(5,q^2)\).     

Cyclic Arcs in PG(2, q)

B.C. Kestenband [9], J.C. Fisher, J.W.P. Hirschfeld, and J.A. Thas [3], E. Boros, and T. Szönyi [1] constructed complete (q ² ? q + l)-arcs in PG(2, q ²), q ≥ 3. One of the interesting properties of these arcs is the fact that they are fixed by a cyclic protective group of order q ² ? q + 1. We investigate the following problem: What are the complete k-arcs in PG(2, q) which are fixed by a cyclic projective group of order k? This article shows that there are essentially three types of those arcs, one of which is the conic in PG(2, q), q odd. For the other two types, concrete examples are given which shows that these types also occur.     

Small semiovals in PG(2, q)

A semioval in a projective plane is a nonempty subset S of points with the property that for every point P ∈ S there exists a unique line ℓ such that . It is known that and both bounds are sharp. We say that S is a small semioval in if . Dover [5] proved that if S has a (q − 1)-secant, then , thus S is small, and if S has more than one (q − 1)-secant, then S can be obtained from a vertexless triangle by removing some subset of points from one side. We generalize this result and prove that if there exist integers 1 ≤ t and − 1 ≤ k such that and S has a (q − t)-secant, then the tangent lines at the points of the (q − t)-secant are concurrent. Specially when t = 1 then S can be obtained from a vertexless triangle by removing some subset of points from one side. The research was supported by the Italian-Hungarian Intergovernmental Scientific and Technological Cooperation Project, Grant No. I-66/99 and by the Hungarian National Foundation for Scientific Research, Grant Nos. T 043556 and T 043758.     

On (q 2 + q + 2, q + 2)-arcs in the Projective Plane PG(2, q)

A (k,n)-arc in PG(2,q) is usually defined to be a set of k points in the plane such that some line meets in n points but such that no line meets in more than n points. There is an extensive literature on the topic of (k,n)-arcs. Here we keep the same definition but allow to be a multiset, that is, permit to contain multiple points. The case k=q ²+q+2 is of interest because it is the first value of k for which a (k,n)-arc must be a multiset. The problem of classifying (q ²+q+2,q+2)-arcs is of importance in coding theory, since it is equivalent to classifying 3-dimensional q-ary error-correcting codes of length q ²+q+2 and minimum distance q ². Indeed, it was the coding theory problem which provided the initial motivation for our study. It turns out that such arcs are surprisingly rich in geometric structure. Here we construct several families of (q ²+q+2,q+2)-arcs as well as obtain some bounds and non-existence results. A complete classification of such arcs seems to be a difficult problem.     

Large blocking sets in PG(2,q2)

Minimal blocking sets in $\mathrm{PG}(2,q^2)$ have size at most $q^3+1$. This result is due to Bruen and Thas and the bound is sharp, sets attaining this bound are called unitals. In this paper, we show that the second largest minimal blocking sets have size at most $q^3+1-(p-3)/2$, if $q=p$, $p\ge 67$, or $q=p^h$, $p>7$, $h>1$. Our proof also works for sets having at least one tangent at each of its points (that is, for tangency sets).     

On regular semiovals in PG(2,q)

In this paper we prove that a point set in PG(2,q) meeting every line in 0, 1 or r points and having a unique tangent at each of its points is either an oval or a unital. This answers a question of Blokhuis and Szőnyi [1]. Research was partially supported by OTKA Grants T 043758, F 043772; the preparation of the final version was supported by OTKA Grant T 049662 and TéT grant E-16/04.     

Tangency sets in PG(3, q)

A tangency set of PG (d,q) is a set Q of points with the property that every point P of Q lies on a hyperplane that meets Q only in P. It is known that a tangency set of PG (3,q) has at most points with equality only if it is an ovoid. We show that a tangency set of PG (3,q) with , or points is contained in an ovoid. This implies the non‐existence of minimal blocking sets of size , , and of with respect to planes in PG (3,q), and implies the extendability of partial 1‐systems of size , , or to 1‐systems on the hyperbolic quadric . © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 462–476, 2008     

Triads, Flocks of Conics and Q -(5,q)

We show that if an ovoid of Q (4,q),q even, admits a flock of conics then that flock must be linear. It follows that an ovoid of PG (3,q),q even, which admits a flock of conics must be an elliptic quadric. This latter result is used to give a characterisation of the classical example Q ^-(5,q) among the generalized quadrangles T ₃( ), where is an ovoid of PG (3q) and q is even, in terms of the geometric configuration of the centres of certain triads.     

On large minimal blocking sets in PG(2,q)

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 q^4/3 + 1 or q^4/3 + 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 . © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 25–41, 2005.