Rigidity Theorems of Hamada and Ohmori, Revisited

摘要

Let Alpha be a (0,1)-matrix of size beta is greater than or equal to upsilon. Suppose that all rows (columns) of Alpha are nonzero and distinct. We show that the rank of Alpha over a field of characteristic 2 satisfies, rank(2) (Alpha) is greater than or equal to log(2) (upsilon + 1), with equality if and only if Alpha is the incidence matrix of a point-hyperplane Hadamard design. This generalizes a rigidity theorem of Hamada and Ohmori, who assumed that upsilon + 1 is a power of 2 and that Alpha is already known to be the incidence matrix of a Hadamard design. Our results follow from a generalization of a rank inequality of Wallis.