Fix a positive integer $\ell$. For $x_1,\dotsc,x_n\in S^{d-1}$, Venkov proved that
$$
\sum_{i=1}^n\sum_{j=1}^n(x_i\cdot x_j)^{2\ell}\geq\frac{(2\ell-1)!!(d-2)!!}{(d+2\ell-2)!!}\cdot n^2,
$$
with equality precisely when $\{x_i\}_{i=1}^n\cup\{-x_i\}_{i=1}^n$ forms a spherical $(2\ell+1)$-design; see Theorem 8.1 in [Réseaux et designs sphériques][1]. Considering Theorem 5.12 in Delsarte, Goethals, and Seidel's [Spherical codes and designs](https://doi.org/10.1007/BF03187604), equality in this lower bound requires $n\geq\binom{d+\ell-1}{\ell}$.
Meanwhile, equality in the simpler lower bound
$$
\sum_{i=1}^n\sum_{j=1}^n(x_i\cdot x_j)^{2\ell}\geq n
$$
is achievable whenever $n\leq d$. When $\ell=1$, the maximum of these two bounds is achievable for every $n$ by taking $\{x_i\}_{i=1}^n$ to be an orthogonal sequence and/or a [tight frame][3]. When $\ell\geq2$, neither bound is achievable when $n$ is between $d$ and $\binom{d+\ell-1}{\ell}$.

**Question:** Are sharper lower bounds known for values of $n$ between $d$ and $\binom{d+\ell-1}{\ell}$?


  [1]: https://jamartin.perso.math.cnrs.fr/Publications/venkovensmath.pdf
  [2]: https://link.springer.com/content/pdf/10.1007%2FBF03187604.pdf
  [3]: https://en.wikipedia.org/wiki/Frame_(linear_algebra)#Tight_frames