The rank of $\mathbb{Z}_{2^s}$-linear Hadamard codes

The canonical $\mathbb{Z}_{2^s}$-linear Hadamard codes are the Gray images of the $\mathbb{Z}_{2^s}$-additive codes. Their rank, the dimension of the binary linear span, was known in closed form only for $s\le 3$, and for $s \ge 4$ 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 $ρ_s(\boldsymbol{k})$ independent of the length. We compute all the coefficients for $s\le 5$, which yields explicit rank formulas over $\mathbb{Z}_4$ and $\mathbb{Z}_8$ (recovering the known ones), and over $\mathbb{Z}_{16}$ and $\mathbb{Z}_{32}$.