Abstract
Combinatorics Block-transitive Steiner t-designs form a central part of the study of highly symmetric combinatorial configurations at the interface of several disciplines, including group theory, geometry, combinatorics, coding and information theory, and cryptography. The main result of the paper settles an important open question: There exist no non-trivial examples with t = 7 (or larger). The proof is based on the classification of the finite 3-homogeneous permutation groups, itself relying on the finite simple group classification.
Highlights
One of the outstanding problems in combinatorial design theory concerns the existence of Steiner t-designs (i.e., t-(v, k, 1) designs) with t > 5
The known examples for t ≤ 5 often encompass a high degree of regularity and establish deep connections to permutation group theory, geometry, combinatorics, coding and information theory, and cryptography
There has been recent progress on the existence problem by characterizing Steiner t-designs which admit a flag-transitive group of automorphisms
Summary
One of the outstanding problems in combinatorial design theory concerns the existence of Steiner t-designs (i.e., t-(v, k, 1) designs) with t > 5. There has been recent progress on the existence problem by characterizing Steiner t-designs which admit a flag-transitive group of automorphisms (cf [20]). Praeger [7] proved the non-existence of block-transitive (Steiner) t-designs for t > 7. They conjectured that there are no non-trivial examples for t = 6. Main Theorem There is no non-trivial Steiner 7-design D admitting a block-transitive group G ≤ Aut(D) of automorphisms. It is based on the classification of the finite 3-homogeneous permutation groups, itself relying on the classification of the finite simple groups. 1365–8050 c 2010 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France
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