Notation: $M$ is a DFA; $L(M)$ is the language accepted by $M$; $\min(M)$ is the minimal automaton equivalent to $M$ derived from a minimization algorithm such as the Hopcroft algorithm; and $|M|$ is the size of $M$: the number of states in $M$.
We are given the alphabet $\{0,1\}$ and some $n \in \mathbb{N}$.
Let's define some sets to set up the question.
$$ A = \{M \mid L(M) \subseteq \{0,1\}^n\}$$
So $A$ is the set of all automata that accept some language whose words are composed of $n$-bit binary strings. My intention here is also to require that all members of $A$ reject strings of other lengths. Building from this, consider
$$ A_{\min} = \{ \min(M) \mid M \in A \} $$
So $A_{\min}$ is the set of all minimized automata from $A$. Building further, let
$$ x = \max \{ |M| \mid M \in A_{\min}\} $$
So $x$ is the size of the largest automaton from $A_{\min}$. Now we can specify the automaton or automata we're looking for:
$$S = \{ M \mid M \in A_{\min} \, \, \text{and} \, \, |M| = x \} $$
How would one go about constructing a member (or members) of $S$? Any will do, but simpler constructions are preferred.
My initial naive attempt to do this failed. I tried building it up by induction, starting from a basis state which had two states: an accepting state and a non-accepting state. For the inductive step, I tried to build a binary tree composed of two different subtrees built from the previous step. This failed because minimization still merged common subtrees together.
