Steiner 2-designs S(2,4,28) with nontrivial automorphisms

In this article designs with parameters S(2,4,28) and nontrivial automorphism groups are classified. A total of 4466 designs were found. Together with some S(2,4,28)\u27s with trivial automorphism groups found by A.Betten, D.Betten and V.D.Tonchev this sums up to 4653 nonisomorphic S(2,4,28) designs

Arcs and Ovals in the Hermitian and Ree Unitals

The hermitian unitals U(q) and the Ree unitals RU(q) are examined for the existence of ovals and arcs. It is shown that U(q) does not have ovals for q > 2 and that RU(q), like U(q), is embedded in a much larger design with block intersections of cardinality ⩽ 2. Arcs of size 3q + 1 are constructed for the Ree unitals RU(q); they are ovals only in the case q = 3. In this case, U(3) and RU(3) are embedded in the same design and its automorphism group, the symplectic group Sp(6, 2), contains the automorphism groups of both the unitals; the coding-theoretic aspects are elucidated

Steiner 2-designs S(2,4,28) with nontrivial automorphisms

In this article designs with parameters S(2,4,28) and nontrivial automorphism groups are classified. A total of 4466 designs were found. Together with some S(2,4,28)\u27s with trivial automorphism groups found by A.Betten, D.Betten and V.D.Tonchev this sums up to 4653 nonisomorphic S(2,4,28) designs

SL(2,q)-Unitals

Unitals of order $n$ are incidence structures consisting of $n^3+1$ points such that each block is incident with $n+1$ points and such that there are unique joining blocks. In the language of designs, a unital of order $n$ is a $2$-$(n^3+1,n+1,1)$ design. An affine unital is obtained from a unital by removing one block and all the points on it. A unital can be obtained from an affine unital via a parallelism on the short blocks. We study so-called (affine) $\mathrm{SL}(2,q)$-unitals, a special construction of (affine) unitals of order $q$ where $q$ is a prime power. We show several results on automorphism groups and translations of those unitals, including a proof that one block is fixed by the full automorphism group under certain conditions. We introduce a new class of parallelisms, occurring in every affine $\mathrm{SL}(2,q)$-unital of odd order. Finally, we present the results of a computer search, including three new affine $\mathrm{SL}(2,8)$-unitals and twelve new $\mathrm{SL}(2,4)$-unitals

Ovoids and spreads of finite classical generalized hexagons and applications

One intuitively describes a generalized hexagon as a point-line geometry full of ordinary hexagons, but containing no ordinary n-gons for n<6. A generalized hexagon has order (s,t) if every point is on t+1 lines and every line contains s+1 points. The main result of my PhD Thesis is the construction of three new examples of distance-2 ovoids (a set of non-collinear points that is uniquely intersected by any chosen line) in H(3) and H(4), where H(q) belongs to a special class of order (q,q) generalized hexagons. One of these examples has lead to the construction of a new infinite class of two-character sets. These in turn give rise to new strongly regular graphs and new two-weight codes, which is why I dedicate a whole chapter on codes arising from small generalized hexagons. By considering the (0,1)-vector space of characteristic functions within H(q), one obtains a one-to-one correspondence between such a code and some substructure of the hexagon. A regular substructure can be viewed as the eigenvector of a certain (0,1)-matrix and the fact that eigenvectors of distinct eigenvalues have to be orthogonal often yields exact values for the intersection number of the according substructures. In my thesis I reveal some unexpected results to this particular technique. Furthermore I classify all distance-2 and -3 ovoids (a maximal set of points mutually at maximal distance) within H(3). As such we obtain a geometrical interpretation of all maximal subgroups of G2(3), a geometric construction of a GAB, the first sporadic examples of ovoid-spread pairings and a transitive 1-system of Q(6,3). Research on derivations of this 1-system was followed by an investigation of common point reguli of different hexagons on the same Q(6,q), with nice applications as a result. Of these, the most important is the alternative construction of the Hölz design and a subdesign. Furthermore we theoretically prove that the Hölz design on 28 points only contains Hermitian and Ree unitals (previously shown by Tonchev by computer). As these Hölz designs are one-point extensions of generalized quadrangles, we dedicate a final chapter to the characterization of the affine extension of H(2) using a combinatorial property