Characterizing Pyramidal Hadamard Designs With the Largest Number of Fixed Points

Abstract

ABSTRACT A symmetric ‐design is said to be ‐pyramidal, with , under the action of a group if acts as an automorphism group fixing points and acting sharply transitively on the remaining ones. We show that, necessarily, . In particular, for a Hadamard design with parameters , for some , it follows that . Recently, working on the complement design, the family of Hadamard ‐designs admitting an abelian ‐pyramidal automorphism group has been completely determined in the case . In this paper, we generalize that result by showing that coincides with the family of Hadamard ‐designs admitting a ‐pyramidal automorphism group, without assuming either that the group is abelian or that is a power of 2.