Uniform tight frames as optimal signals

Non-orthogonal communication is a promising technique for future wireless networks (e.g., 6G and Wi-Fi 7). In the vector channel model, designing efficient non-orthogonal schemes amounts to following extremum problem: \[ \max \min_k \frac{|v_k|^2}{\sigma^2 + \sum_{l \neq k} \langle v_k, v_l \rangle^2} \] where maximum taken among systems $(v_k)_1^N \subset \mathbb{R}^d$ satisfying $c_1 \leq |v_k|^2 c_2$ every $k$, parameter $\sigma>0$ corresponds noise of channel. We show that in case $\sigma = 0$, uniform tight frames are only optimal configurations. also give quantitative bounds on capacity channels with relatively small noise.