Let $G$ be a finite group. We say a subset $A$ of $G$, $|A|=m$, is $(m,i)$-good, $m\geq 1$ and $0\leq i\leq m$, if there exist $g_A\in G$ such that we have $|gA\cap A|=m-i$.

I need some groups such that, for fixed $m$ and $i$, preferably $i$ is much smaller than $m$, the number of $(m, i)$-good subsets is large.

Is this property well known in group theory? Or, is it related to some well-known properties of groups for special values $m$ and $i$?

$\textbf{The motivation for this question:}$ These type of subsets are used to define some special type of participants in secret sharing schemes on groups. Participants are the members of group and subsets with this property can take some special values of share to construct the key.

I do not know how can I upload a large file here. You can download the text file from this address. If it is helpful, I can do this computation for each order of groups.

$\textbf{Addendum 1}:$ http://s8.picofile.com/file/8341377742/Order_8.txt.html

$\textbf{Addendum 2}:$ http://s9.picofile.com/file/8341380550/Order_9.txt.html

$\textbf{Addendum 3}:$ http://s9.picofile.com/file/8341379792/Order_12.txt.html

$\textbf{New question (10/2018)}:$ Is it true that if the $(m,i)$-good numbers of two groups $G$ and $H$ are equal, then $G$ is isomorphic to $H$?

Thanks for helpful answers, references, and comments.