Use of Myhill-Nerode Theorem to prove minimal number of states
You need to find $2^k$ words which belong to different equivalence classes of the Myhill–Nerode relation. Put differently, you need to find a collection of $2^k$ words such that for any two of them $x \neq y$ there is a word $z$ such that $xz \in L$ and $yz \notin L$ or vice versa.