On Maximal Weights of Hadamard Matrices

Let Ωn denote the set of all Hadamard matrices of order n. For H ϵ Ωn, define the weight of H to be the number of 1's in H and is denoted by w(H). For a subset Γ ⊃ Ωn, define the maximal weight of Γ as w(Γ) = max{w(H) | H ϵ Γ}. Two Hadamard matrices are equivalent if one of them can be transformed to the other by permutation and negation of rows and columns, and the equivalence class containing H is denoted by [H]. In this paper, we shall derive lower bounds for w([H]), which are best possible for n ⩽ 20. We shall also determine the exact value of w(Ω32).