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On putative q-analogues of the Fano plane and related combinatorial structures

Title data

Honold, Thomas ; Kiermaier, Michael:
On putative q-analogues of the Fano plane and related combinatorial structures.
In: Hagen, Thomas ; Rupp, Florian ; Scheurle, Jürgen (ed.): Dynamical Systems, Number Theory and Applications : A Festschrift in Honor of Armin Leutbecher’s 80th Birthday. - New Jersey; London; Singapore; Beijing; Shanghai; Hong Kong; Taipei; Chennai; Tokyo : World Scientific , 2016 . - pp. 141-175
ISBN 978-981-4699-86-0
DOI: https://doi.org/10.1142/9789814699877_0008

Abstract in another language

A set ℱq of 3-dimensional subspaces of GF(q)^7, the 7-dimensional vector space over the finite field GF(q), is said to form a q-analogue of the Fano plane if every 2-dimensional subspace of is contained in precisely one member of ℱq. The existence problem for such q-analogues remains unsolved for every single value of q. Here we report on an attempt to construct such q-analogues using ideas from the theory of subspace codes, which were introduced a few years ago by Koetter and Kschischang in their seminal work on error-correction for network coding. Our attempt eventually fails, but it produces the largest subspace codes known so far with the same parameters as a putative q-analogue. In particular we find a ternary subspace code of new record size 6977, and we are able to construct a binary subspace code of the largest currently known size 329 in an entirely computer-free manner.

Further data

Item Type: Article in a book
Refereed: Yes
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II
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: 11 Apr 2016 07:07
Last Modified: 11 Apr 2016 07:07
URI: https://eref.uni-bayreuth.de/id/eprint/32156