__Finite Geometry Seminar__

# Program of the current semester

**This semester the seminar takes place in the usual room 3.607 (ELTE Southern Building).**

## Fourier analysis in the problem of MUBs (mutually unbiased bases) and of the existence of finite projective planes

## Maximum scattered linear sets of \(\mathrm{PG}(1,q^n)\)

In this talk we will investigate maximum scattered linear sets of \(\mathrm{PG}(1,q^n)\).
A point set \(L_U\) of \(\mathrm{PG}(1,q^n)=\mathrm{PG}(W,\mathbb{F}_{q^n})\) is said to be an \(\mathbb{F}_q\)-**linear set of rank** \(k\) if it is
defined by the non-zero vectors of a \(k\)-dimensional \(\mathbb{F}_q\)-vector subspace \(U\) of \(W\)
\[L_U=\{\langle { \mathbf{u}} \rangle_{\mathbb{F}_{q^n}} \colon { \mathbf{u}}\in U\setminus \{{ \mathbf{0}} \}\}.\]
Also, \(L_U\) is said to be **maximum scattered** \(\mathbb{F}_q\)-linear set if \(k=n\) and \(\displaystyle |L_U|=\frac{q^n-1}{q-1}\).
Recently the theory of maximum scattered \(\mathbb{F}_q\)-linear set has increased its importance mainly
because of its connection with MRD-codes, which are intensively studied.
This is a joint work with Bence Csajbók, Giuseppe Marino and Olga Polverino.

## Computer Classification of Hyperovals and KM-arcs

In this talk I will speak about the computer classification of hyperovals and KM-arcs in small Desarguesian planes, i.e. \(\mathrm{PG}(2,q)\). The focus will be on the techniques used in the largest plane where a full classification is known; this is currently \(\mathrm{PG}(2,64)\) for both.

## Geometrical aspects of subspace codes & Applications of projective planes to extremal graph theory

The seminar will consist of two separate 45 minute talks.

**Leo's abstract:** Subspace codes are codes in which the codewords consist of subspaces of a vector space \(V(n,q)\) of dimension \(n\) over the finite field of order \(q\). These codes now receive a lot of attention because of their relevance for transmission of information in wireless networks. Since codewords are subspaces of a vector space \(V(n,q)\), they also can be interpreted as being subspaces of the projective space \(\mathrm{PG}(n-1,q)\) of dimension \(n-1\) over the finite field of order \(q\). This implies that geometrical techniques can be used to construct subspace codes and to investigate properties of subspace codes. In this talk, a number of geometrical results on subspace codes will be presented to show that the theory of subspace codes is a new interesting research domain for finite geometers.

**Sam's abstract:** Projective planes are often studied on their own, but they make a lot of appearances outside finite geometry. The reason is that because of their very symmetrical structure, they are a perfect candidate to consider in graph theoretical problems. Among these, we will focus on problems in extremal graph theory and show how graphs derived from projective planes appear naturally in this context. New results on some invariants of these graphs will be presented. This is joint work with Leo Storme and Francesco Pavese.

## Resolving sets for generalized quadrangles

A resolving set of a graph \(G=(V;E)\) is a subset \(S=(s_1,s_2,...,s_k)\) of vertices such that for each vertex of \(V\), its distance list \((d(v,s_1),d(v,s_2),...,d(v,s_k))\) with respect to \(S\) is unique. A smallest resolving set is called a metric basis and its size is the metric dimension of \(G\). In the talk we consider the metric dimension of the incidence graph of a generalized quadrangle of order \((q,q)\), and prove it to be at least \(\max\{6q-71,4q-7\}\) for any GQ, and at most \(8q\) for the classical GQs (\(W(q)\) and \(Q(4,q)\)). Joint work with Daniele Bartoli, György Kiss and Marcella Takáts.

## Maximum scattered alterek

Legyen \(V=V(r,q^n)\) egy \(\mathrm{GF}(q^n)\)-feletti \(r\)-dimenziós vektortér, \(n,r>1\), és legyen \(U\) a \(V\)-nek egy \(\mathrm{GF}(q)\)-altere. Az \(U\) altér "scattered", ha a \(V\) egydimenziós \(\mathrm{GF}(q^n)\)-altereit legfeljebb egydimenziós \(\mathrm{GF}(q)\)-alterekben metszi. Ha \(rn\) páros, akkor scattered altér rangja maximum \(rn/2\) lehet, az \(rn/2\)-dimenziós scattered altereket "maximum scattered"-nek nevezzük. Előadásomban három módszert mutatok arra, hogyan lehet \(r=2\) esetén eldönteni egy \(n\)-dimenziós altérről, hogy (maximum) scattered-e vagy sem. Könnyű látni, hogy maximum scattered alterek direkt összege is maximum scattered altér, tehát \(V(2,q^n)\)-beli konstrukciókból könnyen lehet magasabb dimenziós példákat gyártani. Az előadás egy Giuseppe Marinoval, Olga Polverinoval és Corrado Zanellaval közös munkán alapul.

## Selectively balancing unit vectors

A set \(U\) of unit vectors is selectively balancing if there are different subsets \(U^+\) and \(U^-\), such that the Euclidean distance between the sum of the vectors in \(U^+\) and in \(U^-\) is less than \(1\). We prove that the minimum number of unit vectors that guarantee a selectively balancing set in \(\mathbb R^n\) is asymptotically \(\frac12n\log n\). Joint work with Hao Chen.

## Resolving sets for higher dimensional projective spaces

Let \(R(n,q)\) be a resolving set for the point-hyparplane incidence graph of \(\mathrm{PG}(n,q)\). In this talk estimates on the size of \(R(n,q)\) are presented. We prove that if \(q\) is large enough then \(|R(n,q)|\geq 2nq-2\frac{n^{n-1}}{(n-2)!}\). This generalizes tha planar result of Héger and Takáts stating that the metric dimension of the point-line incidence graph of a projective plane of order \(q\) is \(4q-4\). Translating the result of Fancsali and Sziklai about higgledy-piggledy lines to the language of resolving sets, we get that If \(q=p^r,\) \(p>n\) and \(q\geq 2n-1\) then \(|R(n,q)|\leq (4n-2)q\). We prove that \(|R(3,q)|\leq 8q\) and \(|R(4,q)|\leq 12q\). In the cases \(p \leq n-1 \) and \(q \leq 2n-2\) we prove \(|R(n,q)| \leq {n^2+n-6}q\). Joint work with Daniele Bartoli, Stefano Marcugini and Fernanda Pambianco.