Given large positive integers $m$ and $n$:
Let $S$ be the set of integers $\{1,2,\dots,m\}$
We are given as input two vectors $L$ and $U$ both over $S^n$ such that:
$$\bigwedge_{i=1}^{n}{L_i \le U_i}$$
We are interested in the set of vectors $X$ over $S^n$ such that:
$$\bigwedge_{i=1}^{n}{L_i \le X_i \le U_i}$$
Notice that if we define $D = U - L$ than there are:
$$\prod_{i=1}^{n}(D_i+1)$$
valid $X$ vectors.
We want an algorithm that takes $L$ and $U$ and generates a self-contained program as output. That program when executed may output any one valid $X$, and be as small as possible (in terms of total code and data).
Therefore we would like to know which valid $X$ vector is the one that is the "most compressible". We can then generate a program that contains that vector compressed in its data section, and the program will simply decompress it and output it.
How can we determine which $X$ is most compressible as a function of $L$ and $U$?
