Residual $q$-Fano Planes and Related Structures
Abstract
One of the most intriguing problems, in -analogs of designs, is the existence question of an infinite family of -analog of Steiner systems, known also as -Steiner systems, (spreads not included) in general, and the existence question for the -analog of the Fano plane, known also as the -Fano plane, in particular. 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, -Steiner systems were found for one set of parameters. In this paper, a definition for the -analog of the residual design is presented. This new definition is different from previous known definition, but its properties reflect better the -analog properties. The existence of a design with the parameters of the residual -Steiner system in general and the residual -Fano plane in particular are examined. We prove the existence of the residual -Fano plane for all , where is a prime power. The constructed structure is just one step from a construction of a -Fano plane.
Keywords
Cite
@article{arxiv.1704.07714,
title = {Residual $q$-Fano Planes and Related Structures},
author = {Tuvi Etzion and Niv Hooker},
journal= {arXiv preprint arXiv:1704.07714},
year = {2017}
}