Block-transitive Steiner @math -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 @math (or larger). The proof is based on the classification of the finite 3-homogeneous permutation groups, itself relying on the finite simple group classification.
The author @cite_10 @cite_13 recently confirmed the non-existence of block-transitive Steiner @math -designs , modulo two special cases that remain elusive.
@cite_13 @cite_10
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This paper takes a significant step towards confirming a long-standing and far-reaching conjecture of Peter J. Cameron and Cheryl E. Praeger. They conjectured in 1993 that there are no non-trivial block-transitive 6-designs. We prove that the Cameron-Praeger conjecture is true for the important case of non-trivial Steiner 6-designs, i.e. for 6-(v,k,@l) designs with @l=1, except possibly when the group is P@CL(2,p^e) with p=2 or 3, and e is an odd prime power. One of the most central and long-standing open questions in combinatorial design theory concerns the existence of Steiner t -designs for large values of t . Although in his classical 1987 paper, L. Teirlinck has shown that non-trivial t -designs exist for all values of t , no non-trivial Steiner t -design with t > 5 has been constructed until now. Understandingly, the case t = 6 has received considerable attention. There has been recent progress concerning the existence of highly symmetric Steiner 6-designs: It is shown in [M. Huber, J. Algebr. Comb. 26 (2007), pp. 453---476] that no non-trivial flag-transitive Steiner 6-design can exist. In this paper, we announce that essentially also no block-transitive Steiner 6-design can exist.