Residual \(q\)-Fano planes and related structures

Tuvi Etzion¹ ; Niv Hooker¹

¹Computer Science Department Technion IIT Haifa 3200003 Israel

The electronic journal of combinatorics, Tome 25 (2018) no. 2

Résumé

One of the most intriguing problems for $q$-analogs of designs, is the existence question of an infinite family of $q$-Steiner systems that are not spreads. In particular the most interesting case is the existence question for the $q$-analog of the Fano plane, known also as the $q$-Fano plane. These questions are in the front line of open problems in block design. There was a common belief and a conjecture that such structures do not exist. Only recently, $q$-Steiner systems were found for one set of parameters. In this paper, a definition for the $q$-analog of the residual design is presented. This new definition is different from previous known definition, but its properties reflect better the $q$-analog properties. The existence of a design with the parameters of the residual $q$-Steiner system in general and the residual $q$-Fano plane in particular are examined. We construct different residual $q$-Fano planes for all $q$, where $q$ is a prime power. The constructed structure is just one step from a construction of a $q$-Fano plane.

Détail

DOI : 10.37236/7107

Classification : 05B25, 51E10
Mots-clés : \(q\)-analog, spreads, \(q\)-Fano plane, \(q\)-Steiner systems, derived design, residual design