87 research outputs found

### A combinatorial characterisation of embedded polar spaces

Some classical polar spaces admit polar spaces of the same rank as embedded
polar spaces (often arisen as the intersection of the polar space with a
non-tangent hyperplane). In this article we look at sets of generators that
behave combinatorially as the set of generators of such an embedded polar
space, and we prove that they are the set of generators of an embedded polar
space

### On the dual code of points and generators on the Hermitian variety H(2n+1,qÂ²)

We study the dual linear code of points and generators on a non-singular Hermitian variety H(2n + 1, q(2)). We improve the earlier results for n = 2, we solve the minimum distance problem for general n, we classify the n smallest types of code words and we characterize the small weight code words as being a linear combination of these n types

### A linear set view on KM-arcs

In this paper, we study KM-arcs of type t, i.e. point sets of size q + t in
PG(2, q) such that every line contains 0, 2 or t of its points. We use field
reduction to give a different point of view on the class of translation arcs.
Starting from a particular F2-linear set, called an i-club, we reconstruct the
projective triads, the translation hyperovals as well as the translation arcs
constructed by Korchmaros-Mazzocca, Gacs-Weiner and Limbupasiriporn. We show
the KM-arcs of type q/4 recently constructed by Vandendriessche are translation
arcs and fit in this family. Finally, we construct a family of KM-arcs of type
q/4. We show that this family, apart from new examples that are not translation
KM-arcs, contains all translation KM-arcs of type q/4

### A new lower bound for the size of an affine blocking set

A blocking set in an affine plane is a set of points $B$ such that every line
contains at least one point of $B$. The best known lower bound for blocking
sets in arbitrary (non-desarguesian) affine planes was derived in the 1980's by
Bruen and Silverman. In this note, we improve on this result by showing that a
blocking set of an affine plane of order $q$, $q\geq 25$, contains at least
$q+\lfloor\sqrt{q}\rfloor+3$ points

### Elation KM-arcs

In this paper, we study KM-arcs in PG(2, q), the Desarguesian projective plane of order q. A KM-arc A of type t is a natural generalisation of a hyperoval: it is a set of q+t points in PG(2, q) such that every line of PG(2, q) meets A in 0, 2 or t points. We study a particular class of KM-arcs, namely, elation KM-arcs. These KM-arcs are highly symmetrical and moreover, many of the known examples are elation KM-arcs. We provide an algebraic framework and show that all elation KM-arcs of type q/4 in PG(2, q) are translation KM-arcs. Using a result of [2], this concludes the classification problem for elation KM-arcs of type q=4. Furthermore, we construct for all q = 2(h), h > 3, an infinite family of elation KM-arcs of type q/8, and for q=2(h), where 4, 6, 7 | h an infinite family of KM-arcs of type q/16. Both families contain new examples of KM-arcs

### The largest Erd\H{o}s-Ko-Rado sets in 2-(v,k,1) designs

An Erd\H{o}s-Ko-Rado set in a block design is a set of pairwise intersecting
blocks. In this article we study Erd\H{o}s-Ko-Rado sets in 2-(v,k,1) designs,
Steiner systems. The Steiner triple systems and other special classes are
treated separately. For unitals we also determine an upper bound on the size of
the second-largest maximal Erdos-Ko-Rado sets

### The small Kakeya sets in $T^{*}_{2}(\mathcal{C})$, $\mathcal{C}$ a conic

A Kakeya set in the linear representation $T^{*}_{2}(\mathcal{C})$,
$\mathcal{C}$ a non-singular conic, is the point set covered by a set of $q+1$
lines, one through each point of $\mathcal{C}$. In this article we classify the
small Kakeya sets in $T^{*}_{2}(\mathcal{C})$. The smallest Kakeya sets have
size $\left\lfloor\frac{3q^{2}+2q}{4}\right\rfloor$, and all Kakeya sets with
weight less than $\left\lfloor\frac{3(q^{2}-1)}{4}\right\rfloor+q$ are
classified: there are approximately $\sqrt{\frac{q}{2}}$ types

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