On α-points of q-analogs of the Fano plane

Titelangaben

Kiermaier, Michael:
On α-points of q-analogs of the Fano plane.
In: Designs, Codes and Cryptography. Bd. 90 (2022) Heft 6 . - S. 1335-1345.
ISSN 1573-7586
DOI: https://doi.org/10.1007/s10623-022-01033-3

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Abstract

Arguably, the most important open problem in the theory of q-analogs of designs is the question regarding the existence of a q-analog D of the Fano plane. As of today, it remains undecided for every single prime power order q of the base field. A point P is called an α-point of D if the derived design of D in P is a geometric spread. In 1996, Simon Thomas has shown that there always exists a non-α-point. For the binary case q = 2, Olof Heden and Papa Sissokho have improved this result in 2016 by showing that the non-α-points must form a blocking set with respect to the hyperplanes. In this article, we show that a hyperplane consisting only of α-points implies the existence of a partition of the symplectic generalized quadrangle W(q) into spreads. As a consequence, the statement of Heden and Sissokho is generalized to all primes q and all even values of q.