On the cardinality of blocking sets in PG(2,q)

Denote by PG(2,q) the finite desarguesian projective plane of order q, where q=p^h, p a prime, q>2. We define the function m(q) as follows: m(q)=q, if q is a square; m(q)=(q+1)/2, if q is a prime; m(q)=p^h–d, if q=p^h with h an odd integer, where d denotes the greatest divisor of h different from h. The following theorem is proved: For any integer k with q+m(q)+1 k q²–m(q), there exists a blocking set in PG(2,q) having exactly k elements.To Professor Adriano Barlotti on his 60th birthday.Research partially supported by G.N.S.A.G.A. (CNR)     

A characterization of exterior lines of certain sets of points in PG (2, q)

Let A and B be disjoint sets of points in PG(2, q) the Desarguesian projective plane of order q, with |A|q, |B|=q+1, such that each line through a point of A meets B (just once). Then B is a line.     

On Sets of Planes in Projective Spaces Intersecting Mutually in One Point

Let be a projective space. In this paper we consider sets of planes of such that any two planes of intersect in exactly one point. Our investigation will lead to a classification of these sets in most cases. There are the following two main results:- If is a set of planes of a projective space intersecting mutually in one point, then the set of intersection points spans a subspace of dimension 6. There are up to isomorphism only three sets where this dimension is 6. These sets are related to the Fano plane.- If is a set of planes of PG(d,q) intersecting mutually in one point, and if q3, 3(q²+q+1), then is either contained in a Klein quadric in PG(5,q), or is a dual partial spread in PG(4,q), or all elements of pass through a common point.     

Results on Maximal Partial Spreads in PG(3, p 3) and on Related Minihypers

This article classifies all {(q + 1), 3, q}-minihypers, small, q = p ^h ₀, h 1, for a prime number p ₀ 7, which arise from a maximal partial spread of deficiency . When q is a third power, the minihyper is the disjoint union of projected PG(5, )'s; when q is a square, also Baer subgeometries PG(3, ) can occur. This leads to a discrete spectrum for the small values of the deficiency of the corresponding maximal partial spreads.     

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 a set of lines of PG(3, q) corresponding to a maximal cap contained in the Klein quadric of PG (5, q)

The problem is considered of constructing a maximal set of lines, with no three in a pencil, in the finite projective geometry PG(3, q) of three dimensions over GF(q). (A pencil is the set of q+1 lines in a plane and passing through a point.) It is found that an orbit of lines of a Singer cycle of PG(3, q) gives a set of size q ³ + q ² + q + 1 which is definitely maximal in the case of q odd. A (q ³ + q ² + q + 1)-cap contained in the hyperbolic (or Klein) quadric of PG(5, q) also comes from the construction. (A k-cap is a set of k points with no three in a line.) This is generalized to give direct constructions of caps in quadrics in PG(5, q). For q odd and greater than 3 these appear to be the largest caps known in PG(5, q). In particular it is shown how to construct directly a large cap contained in the Klein quadric, given an ovoid skew to an elliptic quadric of PG(3, q). Sometimes the cap is also contained in an elliptic quadric of PG(5, q) and this leads to a set of q ³ + q ² + q + 1 lines of PG(3,q ²) contained in the non-singular Hermitian surface such that no three lines pass through a point. These constructions can often be applied to real and complex spaces.     

Some remarks on embeddings of the flag geometries of projective planes in finite projective spaces

The flag geometry of a finite projective plane II of orders is the generalized hexagon of order (s, 1) obtained from II by putting equal to the set of all flags of II, by putting equal to the set of all points and lines of II and where I is the natural incidence relation (inverse containment), that is, is the dual of the double of II in the sense of [8]. Then we say that is fully (and weakly) embedded in the finite projective space PG(d, q) if is a subgeometry of the natural point-line geometry associated with PG(d, q), if s=q, if the set of points of generates PG(d, q) (and if the set of points of not opposite any given point of does not generate PG(d, q)). We have classified all such embeddings in [3, 4, 5, 6]. In the present paper, we weaken the hypotheses in some special cases, and we give an alternative formulation of the classification.     

Proper 2-Covers of PG(3,q), q Even

A k-cover of =PG(3q) is a set S of lines of such that every point is on exactly k lines of S. S is proper if it contains no spread. The existence of proper k-covers of is necessary for the existence of maximal partial packings of q ²+q+1–k spreads of . Here we give the first construction of proper 2-packings of PG(3,q) with q even; for q odd these have been constructed by Ebert.     

Partial t-Spreads in PG(2t+1,q)

This article first of all discusses the problem of the cardinality of maximal partial spreads in PG(3,q), q square, q>4. Let r be an integer such that 2rq+1 and such that every blocking set of PG(2,q) with at most q+r points contains a Baer subplane. If S is a maximal partial spread of PG(3,q) with q ²-1-r lines, then r=s( +1) for an integer s2 and the set of points of PG(3,q) not covered byS is the disjoint union of s Baer subgeometriesPG(3, ). We also discuss maximal partial spreads in PG(3,p ³), p=p ₀ ^h , p ₀ prime, p ₀ 5, h 1, p 5. We show that if p is non-square, then the minimal possible deficiency of such a spread is equal to p ²+p+1, and that if such a maximal partial spread exists, then the set of points of PG(3,p ³) not covered by the lines of the spread is a projected subgeometryPG(5,p) in PG(3,p ³). In PG(3,p ³),p square, for maximal partial spreads of deficiency p ²+p+1, the combined results from the preceding two cases occur. In the final section, we discuss t-spreads in PG(2t+1,q), q square or q a non-square cube power. In the former case, we show that for small deficiencies , the set of holes is a disjoint union of subgeometries PG(2t+1, ), which implies that 0 (mod +1) and, when (2t+1)( -1) <q-1, that 2( +1). In the latter case, the set of holes is the disjoint union of projected subgeometries PG(3t+2, ) and this implies 0 (mod q ^2/3+q ^1/3+1). A more general result is also presented.     

A class of completek-caps inPG(3,q) forq an odd prime

The main problem on caps, posed originally by Segre in the fifties, is to determine the values of k for which there exists a complete k-cap. Very few results on this problem are known. The cardinality of the largest cap(s) and the smallest complete cap(s) are crucial. In this paper it is shown that there exist complete k-caps in PG(3, q), q an odd prime 5 or q = 9, such that k = (q² + q + 6)/3 or k = (q² + 2q + 6)/3. These complete caps are smaller than those currently known for q odd.In memoriam Giuseppe Tallini     

A new graphic characterization of non singular quadrics of PG(r,q)

LetK be ak-set of class [0, 1,m,n]₁ of anr-dimensional projective Galois space PG(r, q) of orderq. We prove that: Ifr = 2s (s 2),k = _2s–1 and if through each point ofK there are exactlyq ^2(s–1) tangent lines and at most _2s–3 n-secant lines, thenK is a non singular quadric of PG(2s,q). Ifr = 2s–1 (s2),k=_2(s–1) +q ^s–1 and if at each point ofK there are exactlyq ^2s–3 –q ^s–2 tangents and at most _2(s–2)+q ^s–2 n-secant lines, thenK is a hyperbolic quadric of PG(2s–1,q).     

Cyclic caps in PG(3,q)

This article investigates cyclic completek-caps in PG(3,q). Namely, the different types of completek-capsK in PG(3,q) stabilized by a cyclic projective groupG of orderk, acting regularly on the points ofK, are determined. We show that in PG(3,q),q even, the elliptic quadric is the only cyclic completek-cap. Forq odd, it is shown that besides the elliptic quadric, there also exist cyclick-caps containingk/2 points of two disjoint elliptic quadrics or two disjoint hyperbolic quadrics and that there exist cyclick-caps stabilized by a transitive cyclic groupG fixing precisely one point and one plane of PG(3,q). Concrete examples of such caps, found using AXIOM and CAYLEY, are presented.     

The Classification of Projective Planes of Order q3 with Cone-Representations in PG (6, q)

The classification of cone-representations of projective planes of orderq ³ of index 3 and rank 4 (and so in PG(6,^q)) is completed. Any projective plane with a non-spread representation (being a cone-representation of the second kind) is a dual generalised Desarguesian translation plane, as found by Jha and Johnson, and conversely. Indeed, given any collineation of PG(2,^q) with no fixed points, there exists such a projective plane of order q³ , where q is a prime power, that has the second kind of cone-representation of index 3 and rank 4 in PG(6,q). An associated semifield plane of order q ³ is also constructed at most points of the plane. Although Jha and Johnson found this plane before, here we can show directly the geometrical connection between these two kinds of planes.     

A Common Characterization of Ovoids and Non-singular Ruled Quadrics in PG(d, n)

A property is given which characterizes the ovoids and the nonsingular ruled quadrics among the nonsingular Tallini sets in PG (d,n), d 2 and n 4.     

Baer subgeometry partitions

For any odd integern 3 and prime powerq, it is known thatPG(n–1, q²) can be partitioned into pairwise disjoint subgeometries isomorphic toPG(n–1, q) by taking point orbits under an appropriate subgroup of a Singer cycle ofPG(n–1, q²). In this paper, we construct Baer subgeometry partitions of these spaces which do not arise in the classical manner. We further illustrate some of the connections between Baer subgeometry partitions and several other areas of combinatorial interest, most notably projective sets and flagtransitive translation planes.     

On the Non-Existence of Certain Cameron-Liebler Line Classes in PG(3, q)

Our main result is a non-existence theorem for certain families of lines in three dimensional projective space PG(3, q) over a finite field GF(q). Specifically, a Cameron-Liebler line class in PG(3, q) is a set of lines which intersects every spread of PG(3, q) in the same number x of lines (this number is called its parameter). These sets arose in connection with an attempt by Cameron and Liebler to determine the subgroups of PGL(n+1, q) which have the same number of orbits on points (of PG(n, q)) as on lines; they satisfy several equivalent properties. Here we prove that for 2 < x q, no Cameron-Liebler line class of parameter x exists in PG(3, q). A relevant general question on incidence matrices is described.     

New Hadamard matrices and conference matrices obtained via Mathon's construction

We give a formulation, via (1, –1) matrices, of Mathon's construction for conference matrices and derive a new family of conference matrices of order 59^2t+1 + 1,t 0. This family produces a new conference matrix of order 3646 and a new Hadamard matrix of order 7292. In addition we construct new families of Hadamard matrices of orders 69^2t+1 + 2, 109^2t+1 + 2, 8499 ^t ,t 0;q ²(q + 3) + 2 whereq 3 (mod 4) is a prime power and 1/2(q + 5) is the order of a skew-Hadamard matrix); (q + 1)q ²9 ^t ,t 0 (whereq 7 (mod 8) is a prime power and 1/2(q + 1) is the order of an Hadamard matrix). We also give new constructions for Hadamard matrices of order 49 ^t 0 and (q + 1)q ² (whereq 3 (mod 4) is a prime power).This work was supported by grants from ARGS and ACRB.Dedicated to the memory of our esteemed friend Ernst Straus.     

On the non-existence of linear codes attaining the Griesmer bound

A lower bound on the size of a set K in PG(3, q) satisfying for any plane of PG(3, q), q4 is given. It induces the non-existence of linear [n,4,n + 1 – q ²]-codes over GF(q) attaining the Griesmer bound for .     

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

Covering the Hecke Triangle Surfaces

In the mid-1980s an equivalence was established between the simple closed geodesics on the Riemann surfaces obtained as quotients of the upper half plane H by any of the following subgroups of the modular group (1) : , (3), and ³. An axis of a hyperbolic element of (1) projects to a simple closed geodesic on one of these surfaces if and only if it does so on the other two.This equivalence was used to obtain a variety of Diophantine and geometric results. In subsequent related investigations, the role of (1) was assumed by the Hecke triangle group G_q for q 3. (For q = 3, we have (1) = G₃.) These works employed the analog of ³, denoted _q.In the context of the G_q, the present paper gives the analog of , which we denote _q. As in the case q = 3, we have [_q:_q] = 2. A rather full discussion of geometry of _q\ H is given. In particular, we demonstrate that the equivalence of simple closed geodesics on _q\ H and _q\ H does not hold for q 7.As of this writing, we have not been able to obtain an appropriate analog of (3).