# Spherical code for interesection of $k$-sparse vectors and unit sphere

Let us assume $$X\in\mathbb{R}^{n\times d}, rank(X)=d$$, integer $$k\in\mathbb{N},k\ll d$$, positive constant $$0<\epsilon<1$$, and $$\mathcal{S}\subset \mathbb{R}^d$$ denotes the unit sphere. We also denote $$k$$-sparse spans of $$X$$ by $$Y = \{X\beta : \beta\in\mathbb{R}^d,|\beta|_0=k\}$$.

We would like to find the largest integer $$m$$ such that there exists vectors $$\{v_1,...,v_m\}\subset (Y\cap\mathcal{S})$$ that satisfiy $$\forall i\neq j\in[m] : \langle v_i,v_j \rangle\ge \epsilon$$, i.e., no pairs of vectors has angle less than $$\cos^{-1}{\epsilon}$$. What is the largest that $$m$$ can be with regard to $$d,n, k$$, and $$\epsilon$$? In other words, which configuration of $$X$$ would lead to the largest possible $$m$$? Asymptotic upper bounds are desirable.

This problem has some resemblance to spherical code. If we only consider vectors on the sphere, it would be spherical code with dimension $$d$$. If we take the intersection of unit sphere with one $$k$$-dimensional hyperplane we get a $$k$$-dimensional unit sphere and the answer would be spherical code for dimension $$k$$. But as we take the intersection with $$\binom{d}{k}$$ hyperplanes of dimension $$k$$, the answer must lie somewhere in between these two.

Is there any reference, book, article that could point to possible answers? Any ideas how to formulate this problem in a way that is more approachable? I know that there are some semi-definite programming approaches to find upper bound for kissing numbers, like this. Could they possibly be adopted for our purposes?