For $q$ the size of the alphabet and $n$ the length of the code it is costumary to denote by $A_q(n,d)$ the maximal size of a code with minimum distance $d$.
There are numerous investigations on this. tergi already mentioned tables of explicit values. There are however also general bounds known. In particular a classical result is the Gilbert-Varshamov bound that says $$A_q (n,d) \ge \frac{q^n}{\sum_{j=0}^{d-1} C(n,j) (q-1)^j}$$ where by $C(n,j)$ I just mean the binomial coefficient but momentarily fail to typeset it properly.
This is not precisely what you need as you have some $x$ given that corresponds to the $A_q(n,d)$ and need to find a suitable $n$. But for concrete values it would now be easy to solve your problem, and if you need explicit bounds they would (with some additional loss) also be obtainable.
Another question would be how to effectively construct the set then. (In principle the proof of the above bound gives a method. But depending on what you are trying to achieve this might not be a good way to proceed.) Yet, without further details from you it is hard/impossible to know what type of information would be most useful.