The rank of Z2ˢ-linear Hadamard codes

The canonical Z2ˢ-linear Hadamard codes are the Gray images of the Z2ˢ-additive codes. Their rank, the dimension of the binary linear span, was known in closed form only for s≤ 3, and for s ≥ 4 only through computer computations for small lengths. We give a method that computes the rank for every s. We show that the rank is a binomial polynomial in the type parameters, with coefficients ρₛ(k) independent of the length. We compute all the coefficients for s≤ 5, which yields explicit rank formulas over Z₄ and Z₈ (recovering the known ones), and over Z₁₆ and Z₃₂.