Notes on constants for maxima of Rademacher averages

Let ${\epsilon }_{ij},i,j\ge 1$ be independent Rademacher variables. We prove \begin{equation*} $\mathbb{E}$ \max_{1\leq j\leq p}\left|\frac{1}{n}\sum_{i=1}^nε_{ij}\right| \geq \min\left{\frac{255}{256},\frac{1}{\sqrt{2\log 2}}\sqrt{\frac{\log(2p)}{n}}\right}. \end{equation*} The equality is attained, for instance, by $(n,p)=(2,1)$ and $(n,p)=(2,8)\mathrm{.}$ We also discuss the optimality of the numerical constants.