The discrepancy of Sylvester-Hadamard matrices in odd dimension

Let $H_k$ be the $2^k\times 2^k$ Sylvester–Hadamard matrix and let $\operatorname{disc}(H_k)$ be the minimum of $\|H_kx\|_\infty$ over sign vectors $x$. It is known that $\operatorname{disc}(H_k)=2^{k/2}$ for even $k$, and it has been conjectured that $\operatorname{disc}(H_k)=2^{(k+1)/2}$ for every odd $k$. We show that $\operatorname{disc}(H_k)=2^k-2\rho(1,k)$, where $\rho(1,k)$ is the covering radius of the first-order Reed–Muller code of length $2^k$. Known values and bounds for this covering radius then show that the conjecture holds for $k\le 7$ and fails for every odd $k\ge 9$; in fact $\operatorname{disc}(H_k)\le \tfrac78\,2^{(k+1)/2}$ for every odd $k\ge 9$. Determining $\operatorname{disc}(H_k)$ for odd $k\ge 9$ is therefore equivalent to determining $\rho(1,k)$.