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

Title data

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

Official URL: Volltext

Abstract in another language

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.

Further data

Item Type: Article in a journal
Refereed: Yes
Keywords: Subspace design; q-analog; Fano plane; Steiner system; Subspace code
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II (Computer Algebra)
Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 10 Sep 2022 21:00
Last Modified: 23 Nov 2022 07:30
URI: https://eref.uni-bayreuth.de/id/eprint/71787