A characterization of Buekenhout-Metz unitals in PG(2, q 2), q even

The known examples of embedded unitals (i.e. Hermitian arcs) in PG(2, q ²) are B-unitals, i.e. they can be obtained from ovoids of PG(3, q) by a method due to Buekenhout. B-unitals arising from elliptic quadrics are called BM-unitals. Recently, BM-unitals have been classified and their collineation groups have been investigated. A new characterization is given in this paper. We also compute the linear collineation group fixing the B-unital arising from the Segre-Tits ovoid of PG(3, 2 ^r ), r3 odd. It turns out that this group is an Abelian group of order q ².Research supported by MURST.     

A condition for the existence of ovals in PG(2, q), q even

A condition is found that determines whether a polynomial over GF(q) gives an oval in PG(2, q), q even. This shows that the set of all ovals of PG(2, q) corresponds to a certain variety of points of PG((q–4)/2, q). The condition improves upon that of Segre and Bartocci, who proved that all the terms of an oval polynomial have even degree. It is suitable for efficient computer searches.     

Small complete caps in PG(N,q), q even

A new family of small complete caps in PG(N,q), q even, is constructed. Apart from small values of either N or q, it provides an improvement on the currently known upper bounds on the size of the smallest complete cap in PG(N,q): for N even, the leading term is replaced by with , for N odd the leading term is replaced by with . © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 420–436, 2007     

Constructing some designs invariant under PSL2(q), q even

In this paper, we use Key-Moori methods 1 and 2 to construct some designs from the maximal subgroups and conjugacy classes of the group PSL₂(q), where q is a power of 2.     

A characterization of PGL(2,q),q ODD

LetG be a finite group andA andB solvable subgroups ofG, such thatG=AB and 2 is the only common prime divisor ofA andB. Under suitable restrictions of the 2-structure ofA andB, it is shown that eitherG is solvable orG contains a solvable normal subgroupN, such thatG/N contains a normal subgroup, which is isomorphic to PGL(2,q),q odd.     

A characterization of PGL(2, q), q odd

Following the lines of [10], we give a characterization of the group PGL(2, q), q odd, in terms of involutions.Work performed under the auspicies of G.N.S.A.C.A. of C.N.R. supported by the 40% grants of M.P.I.     

On Buekenhout-Metz unitals in PG(2,q 2),q even

Research partially supported by M.P.I. (Research project Strutture geometriche combinatorie e loro applicazioni).     

A characterization of the generalized quadrangle Q(4,q), q ODD

In [3], [ 4 ] we introduced the concept of (0,2)-set in generalized quadrangles, in order to obtain characterizations for P(S,()) and T ₂ ^* (O). Using these sets we are now able to formulate a characterization for Q(4,q), q odd, by assuming local conditions in an antiregular point x of a generalized quadrangle of order s.     

Generalized quadrangles of order (q,q 2),q even,containingW(q) as a subquadrangle

In this paper we give a characterization of the generalized quadrangleQ(5,q),q even, in terms of ovoids of its subquadrangles.     

Caps in PG(n,q),q even,n⩾3

Letm′₂(3,q) be the largest value ofk(k<q ²+1) for which there exists a completek-cap in PG(3,q),q even. In this paper, the known upper bound onm′₂(3,q) is improved. We also describe a number of intervals, fork, for which there does not exist a completek-cap in PG(3,q),q even. These results are then used to improve the known upper bounds on the number of points of a cap in PG(n, q),q even,n?4.     

New inductive constructions of complete caps in PG(N,q), q even

Some new families of small complete caps in PG(N, q), q even, are described. By using inductive arguments, the problem of the construction of small complete caps in projective spaces of arbitrary dimensions is reduced to the same problem in the plane. The caps constructed in this article provide an improvement on the currently known upper bounds on the size of the smallest complete cap in PG(N, q), N≥4, for all q≥2³. In particular, substantial improvements are obtained for infinite values of q square, including q=2^2Cm, C≥5, m≥3; for q=2^Cm, C≥5, m≥9, with C, m odd; and for all q≤2¹⁸. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 177–201, 2010     

On the smallest non-trivial tight sets in Hermitian polar spaces H(d,q2), d even

We show that every $x$-tight set of a Hermitian polar spaces $\mathrm{H}(2n,q^2)$, $n\ge 2$, is the union of $x$ disjoint generators of the polar space provided that $x\le \frac{1}{2}(q+1)$. This was known before only when $n\in \{2,3\}$. This result is a contribution to the conjecture that the smallest $x$-tight set of $H(2n,q^2)$ that is not a union of disjoint generators occurs for $x=q+1$ and is for sufficiently large $q$ an embedded subgeometry.     

A characterization of Grassmann graphs

Let q be a prime power and suppose that e and n are integers satisfying 1 e n − 1. Then the Grassmann graph Γ(e, q, n) has as vertices the e-dimensional subspaces of a vector space of dimension n over the field F_q, where two vertices are adjacent iff they meet in a subspace of dimension e − 1. In this paper, a characterization of Γ(e, q, n) in terms of parameters is obtained provided that and ( if q ε {2, 3}) and if q = 3). As a consequence we can show that these Grassmann graphs are uniquely determined as distance-regular graphs by their intersection arrays.     

Subschemes of Hamming association schemes H(n,q), q≥4

The subschemes of the Hamming schemes H(n, q) for q greater than or equal to 4 are studied. We prove that there are no nontrivial subschemes when q > 4 and there exists only one nontrivial subscheme when q = 4.