We are given a set of $w$ values, i.e. $\{w_1, w_2, \dots, w_n\}$, and a set of $v$ values $\{v_1, v_2, \dots, v_m\}$. The question is really simply, what is the (average) sum of minimum powers of $v$'s to represent all of the $w$'s, modulo $q$?
In other words, if $w_1 = (v_2)^{10} * (v_3)^3$, then the sum of powers to represent $w_1$ is $10+3=13$. I'm looking for the minimum sum of powers for each $w$. Then we total the minimum sum of powers for all $w$s, and this gives us a value. Then, again, I'm looking for the value that I can expect, on average, for a given $m$, $n$, and $q$.
OBSERVATIONS
I've plotted the minimum sum for $3$, $4$, and $5$ $w$s, modulo $37$, as shown below. The horizontal axis is the total size of the powers:
I also plotted the minimum sum for $4$ values modulo $37$, using $3$ separate rounds consisting of $1000$ trials each, as shown here:
