JournalspmVol. 74, No. 3pp. 213–232

Synchronization and separation in the Johnson schemes

  • Mohammed Aljohani

    University of St Andrews, UK
  • John Bamberg

    University of Western Australia, Perth, Australia
  • Peter J. Cameron

    University of St Andrews, UK
Synchronization and separation in the Johnson schemes cover
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Abstract

Recently Peter Keevash solved asymptotically the existence question for Steiner systems by showing that S(t,k,n)S(t,k,n) exists whenever the necessary divisibility conditions on the parameters are satisfied and nn is sufficiently large in terms of kk and tt. The purpose of this paper is to make a conjecture which if true would be a significant extension of Keevash’s theorem, and to give some theoretical and computational evidence for the conjecture.

We phrase the conjecture in terms of the notions (which we define here) of synchronization and separation for association schemes. These definitions are based on those for permutation groups which grow out of the theory of synchronization in finite automata. In this theory, two classes of permutation groups (called synchronizing and separating) lying between primitive and 22-homogeneous are defined. A big open question is how the permutation group induced by SnS_n on kk-subsets of {1,,n}\{1,\ldots,n\} fits in this hierarchy; our conjecture would give a solution to this problem for nn large in terms of kk. We prove the conjecture in the case k=4k=4: our result asserts that SnS_n acting on 44-sets is separating for n10n\ge10 (it fails to be synchronizing for n=9n=9).

Cite this article

Mohammed Aljohani, John Bamberg, Peter J. Cameron, Synchronization and separation in the Johnson schemes. Port. Math. 74 (2017), no. 3, pp. 213–232

DOI 10.4171/PM/2003