In a $d$ dimensional vector space defined over $\mathbb{C}$, how do I calculate the largest number $N(\epsilon, d)$ of vectors $\{V\_i\}$ which satisfies the following properties. Here $\epsilon$ is smaller but finite compared to 1.


$$\langle V\_i, V\_i\rangle = 1$$


$$|\langle V\_i, V\_j\rangle| \leq \epsilon, i \neq j$$


Some examples are as follows. 


1. $N(0, d)$ = d
2. $N\left(\frac{1}{2}, 2\right)$ = 3
3. $N\left(\frac{1}{\sqrt{2}}, 2\right) = 6$


How do I obtain any general formula for $N(\epsilon, d)$. Even an approximate form for $N(\epsilon, d)$ in the large $d$ and small $\epsilon$ ($\epsilon \ll 1$) limit works fine for me.



Neat/Approximate formula for maximum number of "almost orthogonal" vectors in a complex vector space?
The variant of the Johnson-Lindenstrauss lemma that you can use is derived by L. Welch in [Lower bounds on the maximum cross correlation of signals](https://ieeexplore.ieee.org/document/1055219)  (1974). This paper is behind a paywall, I quote the result from [arXiv:0909.0206](https://arxiv.org/abs/0909.0206)


Consider $N=d^{k}$ unit vectors $V\_i$ in $\mathbb{C}^d$ with $N>d$. Then the maximal inner product $\epsilon=\max\_{i\neq j}|\langle V\_i|V\_j\rangle|$ satisfies the inequality
$$\epsilon^{2k}\geq \frac{1}{N-1}\left(\frac{N}{{{d+k-1}\choose{k}}}-1\right).$$


