A note on representing dowling geometries by partitions

František Matúš; Aner Ben-Efraim

Kybernetika (2020)

  • Volume: 56, Issue: 5, page 934-947
  • ISSN: 0023-5954

Abstract

top
We prove that a rank ≥ 3 Dowling geometry of a group H is partition representable if and only if H is a Frobenius complement. This implies that Dowling group geometries are secret-sharing if and only if they are multilinearly representable.

How to cite

top

Matúš, František, and Ben-Efraim, Aner. "A note on representing dowling geometries by partitions." Kybernetika 56.5 (2020): 934-947. <http://eudml.org/doc/297015>.

@article{Matúš2020,
abstract = {We prove that a rank $\ge 3$ Dowling geometry of a group $H$ is partition representable if and only if $H$ is a Frobenius complement. This implies that Dowling group geometries are secret-sharing if and only if they are multilinearly representable.},
author = {Matúš, František, Ben-Efraim, Aner},
journal = {Kybernetika},
keywords = {matroid representations; partition representations; Dowling geometries; Frobenius groups},
language = {eng},
number = {5},
pages = {934-947},
publisher = {Institute of Information Theory and Automation AS CR},
title = {A note on representing dowling geometries by partitions},
url = {http://eudml.org/doc/297015},
volume = {56},
year = {2020},
}

TY - JOUR
AU - Matúš, František
AU - Ben-Efraim, Aner
TI - A note on representing dowling geometries by partitions
JO - Kybernetika
PY - 2020
PB - Institute of Information Theory and Automation AS CR
VL - 56
IS - 5
SP - 934
EP - 947
AB - We prove that a rank $\ge 3$ Dowling geometry of a group $H$ is partition representable if and only if $H$ is a Frobenius complement. This implies that Dowling group geometries are secret-sharing if and only if they are multilinearly representable.
LA - eng
KW - matroid representations; partition representations; Dowling geometries; Frobenius groups
UR - http://eudml.org/doc/297015
ER -

References

top
  1. Beimel, A., Ben-Efraim, A., Padró, C., Tyomkin, I., 10.1007/978-3-642-54242-8_17, Theory Cryptogr. Conf. 14 (2014), 394-418. MR3183548DOI10.1007/978-3-642-54242-8_17
  2. Ben-Efraim, A., 10.1016/j.disc.2016.02.012, Discrete Math. 339 (2015), 8, 2136-2145. MR3500143DOI10.1016/j.disc.2016.02.012
  3. Brickell, E. F., Davenport, D. M., On the classification of ideal secret sharing schemes. 
  4. Brown, R., 10.1090/memo/0717, Memoirs Amer. Math. Soc. 151 (2001), 717. MR1828640DOI10.1090/memo/0717
  5. Dowling, T. A., 10.1016/s0095-8956(73)80007-3, J. Combinat. Theory, Ser. B 14 (1973), 61-86. MR0307951DOI10.1016/s0095-8956(73)80007-3
  6. Evans, D. M., Hrushovski, E., 10.1112/plms/s3-62.1.1, Proc. London Math. Soc. 62 (1989), 3, 1-24. MR1078211DOI10.1112/plms/s3-62.1.1
  7. Feit, W., Characters of Finite Groups., W. A. Benjamin Company, Inc., New York 1967. MR0219636
  8. Matúš, F., 10.1016/s0012-365x(99)00004-7, Discrete Math. 203 (1999), 169-194. MR1696241DOI10.1016/s0012-365x(99)00004-7
  9. Jacobson, N., Basic Algebra II. (Second Edition), W. H. Freeman and Co., New York 1989. MR1009787
  10. Oxley, J. G., 10.1093/acprof:oso/9780198566946.001.0001, Oxford University Press Inc., New York 2011. MR2849819DOI10.1093/acprof:oso/9780198566946.001.0001
  11. Passman, D. S., Permutation Groups., Dover Publications, Inc. Mineola, New York 2012. MR2963408
  12. Pendavingh, R. A., Zwam, S. H. M. van, 10.1016/j.aam.2011.08.003, Adv. Appl. Math. 50 (2013), 1, 201-227. MR2996392DOI10.1016/j.aam.2011.08.003
  13. Seymour, P. D., 10.1016/0095-8956(92)90007-k, J. Combinat. Theory, Ser. B 56 (1992), 69-73. MR1182458DOI10.1016/0095-8956(92)90007-k
  14. Simonis, J., Ashikhmin, A., 10.1023/a:1008244215660, Designs Codes Cryptogr. 14 (1998), 2, 179-197. MR1614357DOI10.1023/a:1008244215660
  15. Suzuki, M., Group Theory I., Springer-Verlag, Berlin 1982. MR0648772
  16. Vertigan, D., 10.1007/s00026-015-0250-4, Ann. Combinat. 19 (2015), 225-233. MR3319870DOI10.1007/s00026-015-0250-4

NotesEmbed ?

top

You must be logged in to post comments.