An h by h matrix with entries ±1 is called a modular Hadamard matrix if the inner product of any two distinct row vectors is a multiple of a fixed (positive) integer n; such a matrix is also referred to as an “H(n, h) matrix” with parameters n and h. Modular Hadamard matrices and the related combinatorial designs were introduced in [2]; that paper was concerned mainly with two of the related designs, the “pseudo (ν, k, λ)- designs” and the “ (m, v, k1, λ1, k2, λ2, f, λ3)-designs” (the reader is referred to [2] for the definition of these designs).