Derived designs of q-Fano planes and q-analogs of group divisible designs

Titelangaben

Kiermaier, Michael ; Kurz, Sascha ; Wassermann, Alfred:
Derived designs of q-Fano planes and q-analogs of group divisible designs.
2019
Veranstaltung: Academy Contact Forum: Coding Theory and Cryptography VIII , 2019-09-27 , The royal Flemish academy of Belgium for science and arts, Brussels, Belgium.
(Veranstaltungsbeitrag: Kongress/Konferenz/Symposium/Tagung , Vortrag )

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Abstract

Arguably, the most important open problem in the theory of q-analogs of combinatorial designs is the question for the existence of a q-analog of the Fano plane, which is a 2-(7,3,1)_q subspace design. It is open for any single prime power q ≥ 2. Geometrically, a q-analog of the Fano plane is a set F of planes in \operatorname{PG}(6,q) covering every line precisely once.

One approach is to investigate the image of F modulo a fixed point P. It consists of a set L of lines and a set B of planes (which are the images of the blocks of F containing P and not containing P, respectively).

The set L is the derived design of F with respect to P. It is a 1-(6,2,1)_q design or, in other words, a line spread in PG(5,q). If it is a Desarguesian spread, P is called an α-point. It has been shown by Simon Thomas that there must be a point which is not an α-point. For q = 2 this result has been strengthened in 2016 by Heden and Sissokho to the statement that every hyperplane of PG(6,q) contains a non-α-point. We will investigate this question for general q. As a result, the existence of a hyperplane of PG(6,q) consisting only of α-points will imply the existence of a partition of a linear complex in PG(3,q) into q+1 line spreads.

The set B forms a q-analog of a group divisible design (q-GDD) with respect to the line spread L. Its parameters are (6,3,q^2,2)_q, meaning that the planes in B are disjoint to all lines in L and cover all the other lines of PG(5,2) exactly q^2 times. The existence of such q-GDDs follows from a result of Etzion and Hooker in 2018. We will investigate q-GDDs in general and give a construction of an infinite family of q-GDDs which includes the above mentioned parameters.