Bussey systems and Steiner's tactical problem

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.