Designs, permutations, and projective planes

A notion of $t$-designs in the symmetric group on $n$ letters was introduced by Godsil in 1988. In particular $t$-transitive sets of permutations form a $t$-design. We derive lower bounds for $t=1$ and $t=2$ by a power moment method. For general $n,$ and $t\le n/2,$ we give a linear programming lower bound involving Charlier polynomials. For specific $n$ and $t=2,$ this bound is strong enough to show that sharply transitive $2$-designs do not exist for many composite integers $n.$ This implies the non-existence of projective planes of order $10,15,18,24,26,28,33,35.$ In general, based on numerical evidence, we conjecture that $t$-designs have size $n(n-1)\dots (n-t+1),$ or larger.