Given the Hadamard matrix $H_k \in \{ \pm 1\}^{2^k}$ constructed recursively as $\begin{bmatrix} H_{k-1} & H_{k-1} \\ H_{k-1} & - H_{k-1} \end{bmatrix}$ with $H_0 = [1]$, find a vector in $\{ -1,0,1 \}^{2^k}$ which maximizes $\| H_k v\|_1$? I am also interested in the value achieved by the maximizer itself.
A naive upper bound shows that $\| H_k v \|_1 \le \sqrt{2^k} \| H_k v \|_2 = 2^k \| v \|_2 \le (2^k)^{3/2}$. I wonder if it is achievable given the integrality constraint on the vector $v$.