Define $\;\; f \: : \: \{0,1,2,3,...\} \: \to \: \{0,1,2,3,...\} \;\;$ by having $\:f(n)$ be
the number of elements in the largest subsets $\mathbf{S}$ of $\:2^{\{0,1,2,...,\hspace{.01 in}n-1\}}$ such that
for all members $\:X,Y,Z\:$ of $\hspace{.02 in}\mathbf{S}$, $\;\;\;$ if $\;\; X$$\Delta$$ Y \: \subseteq \: Z \;\;$ then $ \; X=Y \;\;\;\; $ .
I can see that $\: \hspace{.01 in}f(0) = 1 = \hspace{.01 in}f(1) \:$ and $\: \hspace{.01 in}f(2) = 2 \:$ and $\: \hspace{.01 in}f(3) = 3 \:$ and
for all non-negative integers $m$ and $n$,
$f(m) \cdot f(n) \: \leq \: \hspace{.01 in}f(m+n) \;\;$ and $\;\; f(n) \leq \hspace{.02 in}\max\left(1,2^n-2\right) \;\;$.
It follows that, for all non-negative integers $n$, $\;\; 2^{\lfloor n/3\rfloor} \leq \hspace{.01 in}f(n) \:\:\:$.
What more can be said about $\:f\hspace{.015 in}$? $\;\;$ I'm particularly interested in lower bounds on $\:f$.
(The application is in cryptography, although the statement doesn't involve crypto.)