I want to maximize the pairwise Frobenius distance between $M$ orthogonal matrices. That is, I'm looking for $Q_{i}, i = 1, 2, ... M$ such that \begin{equation*} \begin{aligned} & \underset{ 1 \leq i, j \leq M}{\text{min}} & & || Q_{i} - Q_{j} ||^2_F \end{aligned} \end{equation*} is maximized. I know there are numerical algorithms that can provide the answer to that. I'm just wondering if the solution has a simple structure. For example, if the solution can be parameterized by a single variable $t$ (and possibly some orthogonal matrix $A$).
Edit: It would also be nice if I someone could lead me to directions on how to bound that.