I am interested in the existence of a set of vectors $\{ v_{ij} \}_{ij} \subseteq \mathbb{C}^N$ for $i \in \{1,\dots,N\}$, $j \in \{1,\dots,N+1\}$ such that $\left\vert v^*_{ij} v_{ij'} \right\vert = 1/N$ for $j \ne j'$ and $v^*_{ij} v_{i'j} = \delta_{ii'}$.

For real vectors, the condition $\left\vert v^*_{ij} v_{ij'} \right\vert = 1/N$ is satisfied if $\{ v_{ij} : i \}$ is a regular simplex for each $j$. I have verified numerically the existence of such solutions for $N \le 5$. For $N=3$ a solution is given by the Compound of three tetrahedra.

For $N=p^k$ with $p$ prime, a solution is given by a set of mutually unbiased bases. These are not known to exist when $N \ne p^k$ but my conditions are looser than the MUB conditions.

Numerical experiments show that such vectors exist for $N \le 6$. Do they exist for all $N$?