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 special lower bounds for $t=1$ and $t=2$ by a power moment method. For general $n,t$ we give a %linear programming lower bound . For $n\ge 4$ and $t=2,$ this bound is strong enough to show a lower bound on the size of such $t$-designs of $n(n-1)\dots (n-t+1),$ which is best possible when sharply $t$-transitive sets of permutations exist. This shows, in particular, that tight $2$-designs do not exist.

翻译：Godsil于1988年引入了$n$元对称群中$t$-设计的概念。特别地，$t$-传递置换集构成一个$t$-设计。我们通过幂矩方法推导出$t=1$和$t=2$的特殊下界。对于一般的$n,t$，我们给出了一个线性规划下界。当$n\ge 4$且$t=2$时，该下界足以证明此类$t$-设计的大小至少为$n(n-1)\dots (n-t+1)$，且在存在锐$t$-传递置换集时为最优。这特别表明，紧$2$-设计并不存在。