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Glasnik matematički, Vol. 58 No. 2, 2023.

Original scientific paper

https://doi.org/10.3336/gm.58.2.04

Bussey systems and Steiner's tactical problem

Charles J Colbourn orcid id orcid.org/0000-0002-3104-9515 ; Arizona State University, Tempe, Arizona, USA
Donald L Kreher orcid id orcid.org/0000-0002-5031-9410 ; Michigan Technological University, Houghton, Michigan, USA
Patric R. J Ostergård orcid id orcid.org/0000-0003-0426-9771 ; Department of Information and Communications Engineering, Aalto University School of Electrical Engineering, 00076 Aalto, Finland

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Abstract

In 1853, Steiner posed a number of combinatorial (tactical) problems, which eventually led to a large body of research on Steiner systems.
However, solutions to Steiner's questions coincide with Steiner systems only for strengths two and three.
For larger strengths, essentially only one class of solutions to Steiner's tactical problems is known, found by Bussey more than a century ago.
In this paper, the relationships among Steiner systems, perfect binary one-error-correcting codes, and solutions to Steiner's tactical problem (Bussey systems) are discussed.
For the latter, computational results are provided for at most 15 points.

Keywords

Steiner system, original Steiner system, Bussey system, perfect code

Hrčak ID:

312011

URI

https://hrcak.srce.hr/312011

Publication date:

15.5.2024.

Visits: 138 *

Publication date: 27 December 2023

Volume: Vol 58

Issue: Svezak 2

Pages: 201-224

DOI: 10.3336/gm.58.2.04

Article Information (continued)

Categories:

Subject: Izvorni znanstveni članak

Keywords:

Keyword: Steiner system, original Steiner system, Bussey system, perfect code

Bussey systems and Steiner's tactical problem

Charles J Colbourn[1]

Email: Charles.Colbourn@asu.edu

Donald L Kreher[2]

Email: Kreher@mtu.edu

Patric R. J Ostergård[3]

Email: patric.ostergard@aalto.fi

 

[1] Arizona State University, Tempe, Arizona, USA

[2] Michigan Technological University, Houghton, Michigan, USA

[3] Department of Information and Communications Engineering, Aalto University School of Electrical Engineering, 00076 Aalto, Finland

Abstract

In 1853, Steiner posed a number of combinatorial (tactical) problems, which eventually led to a large body of research on Steiner systems. However, solutions to Steiner's questions coincide with Steiner systems only for strengths two and three. For larger strengths, essentially only one class of solutions to Steiner's tactical problems is known, found by Bussey more than a century ago. In this paper, the relationships among Steiner systems, perfect binary one-error-correcting codes, and solutions to Steiner's tactical problem (Bussey systems) are discussed. For the latter, computational results are provided for at most 15 points.

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