Extremal solutions of an inequality concerning supports of permutation groups and punctured Hadamard codes

Abstract

If $S$ is the degree of a permutation group and $s$ is the maximum degree of its elements, then $S\le 2s-2$. We show that this inequality is sharp for some permutation group if and only if $s$ is a power of $2$, and then there is exactly one such permutation group up to isomorphism. The unique example is an elementary Abelian $2$-group that arises from a punctured Hadamard code. Then we discuss the solutions of $S=2s-3$ and $S=2s-4$.