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 communication schemes amounts to the following extremum problem: $$\max \min_k \frac{|v_k|^2}{\sigma^2 + \sum_{l \neq k} \langle v_k, v_l \rangle^2}$$ where the maximum is taken among vector systems $(v_k)_1^N \subset \mathbb{R}^d$ satisfying $c_1 \leq |v_k|^2 \leq c_2$ for every $k$, and the parameter $\sigma>0$ corresponds to the noise of the channel. We show that in the case $\sigma = 0$, uniform tight frames are the only optimal configurations. We also give quantitative bounds on the optimal capacity of vector channels with relatively small noise.