Packing problem over discrete space Let $q$ be an positive integer and $F = \{0,1,\dots,q-1\}$, we define Hamming distance in $F^n$ between $x = (x_1, \dots, x_n), y = (y_1, \dots, y_n) \in F^n$ is the number of indices $i$ such that $x_i \neq y_i$.
Question. For $R, D > 0$. What is maximum number of non-overlap circles of radius $D$ can be packed in to a circle of radius $R$?
For a fixed point $x \in F^n$. The number of point in $C(x,R)$ - circle of radius $R$ with center $x$ - is
$$V(n,R) = \sum_{i = 0}^R \binom{n}{i} (q-1)^i.$$
Therefore, we have a trivial upper bound for this question is $\frac{V(n,R)}{V(n,D)}$. Any better bound we can have?
 A: Edit:
The question stated packing in an arbitrary sphere of radius $R,$ which can be less than $n.$ The code examples below being cyclic have a homegeneity property on coordinates. They also support designs and orthogonal arrays. This may be a fruitful direction to investigate the packing radius of the projections of these codes onto $R$ coordinates and look at the worst case packing radius.
Original Answer:
This is a question about coding theory.
If the upper bound you mention is attained, we have what is called a perfect code; your  bound is known as the Hamming bound in coding theory. I will use $t$ for the packing radius, $D$ is used for the Hamming distance normally.
For $t=1,$ the whole space made up of all words $F^n$ is a trivial perfect code.
Other trivial perfect codes are the repetition code of odd length $n$ when $q=2,$ namely
$$
\{ (0,0,\ldots,0),(1,1,\ldots,1)\}
$$
which has $t=n.$ It has been proved that the only other perfect codes that exist when $q=p^m,$ i.e., a power of a prime are the Hamming code $(q=2,n=2^m-1,t=1),$ and the binary $(q=2,n=23,t=3)$ and ternary $(q=3,n=11,t=2),$ Golay codes.
No other perfect codes are known.
When a perfect code doesn't exist there are some lower bounds on $t,$ including the Gilbert-Varshamov bound which guarantees the existence of a code with a given $t,$ once $q,n$ are specified. There are other results as well, too complicated to summarize here. The big textbook at graduate level is Conway and Sloane's Sphere Packings book.
https://link.springer.com/book/10.1007/978-1-4757-6568-7
Depending on your background, I suggest starting with Jon Hall's coding theory text which is available online for a gentler introduction to the topic of coding theory:
https://users.math.msu.edu/users/halljo/classes/codenotes/coding-notes.html
This chapter covers some bounds:
https://courses.cs.washington.edu/courses/cse533/06au/lecnotes/lecture5.pdf
See also below:
https://en.wikipedia.org/wiki/Hamming_bound
https://en.wikipedia.org/wiki/Gilbert%E2%80%93Varshamov_bound
https://en.wikipedia.org/wiki/Golay_code
