I assume that this sort of question has already been considered at great length. Nevertheless, I could not find an answer to this question in the related literature.


Given a constant $a\in (0,2]$, what are the known upper bounds of the maximum number of points we can place on the $d$-dimensional unit sphere such that the Euclidean distance between any two of them is larger or equal to $a$, for $d\gg 1$?



Upper bounds for high-dimensional spherical codes given the covering radius
If I interpreted your question correctly this is really about *spherical codes*. This is a difficult problem that is unsolved in general, there are some bounds, however.


I will use $n$ for the dimension, since $d$ is used for minimum distance of related binary codes.


A spherical code is a collection of points on the surface of the $n-$ sphere. There are some known constructions and a series of bounds. I recommend pages 24-29 of Conway and Sloane's *Sphere Packings, Lattices and Groups*, as a good place to get started with plenty of references.


**Link with coding theory:** If we have a binary code (a subset of $\{0,1\}^n$) of length $n$ and minimum hamming distance $d,$ we can map (say) the symbol $0$ to $+1$ and the symbol $1$ to $-1$ and normalize by $\sqrt{n}$ to obtain points on the Euclidean unit sphere, i.e., a spherical code, where the *maximal angle* between any two codewords is $\phi=\cos^{-1}\left(1-\frac{2d}{n}\right)$. And this gives a Euclidean minimum distance of $$2 \sin(\phi/2)$$ for the spherical code. This, however, does not usually give the best constructions since there aren't that many good pairwise constant distance codes.


**Bounds:** Let $A(n,\phi)$ be the largest possible number of points on the $n-$dimensional unit sphere with maximal angle $\leq \phi,$ then some lower bounds on $A(n,\phi)$ are known. In general, for large $n$ is the nonconstructive bound via random coding arguments dating back to Shannon:


$$A(n,\phi)\geq 2^{-n \log\_2 sin \phi (1+o(1))}.$$


Other "nice" bounds include
$$A(n,\phi)=n+1,\quad for \quad \frac{\pi}{2} \leq \phi \leq \sec^{-1}(-n),$$
for a regular simplex,
$$A(n,\pi/2)=2n,$$
for a regular cross-polytope.


Also look up the Kabatianskii-Levenshtein bound.


I know there is much more recent work on Spherical codes by H. Cohn and others but I have not kept up with developments.


**Edit:** The recent paper by Manin and Marcolli available [here](https://authors.library.caltech.edu/97055/2/1801.01552.pdf) might help.


