Find binary number with max hamming distance wrt given set of binary numbers

Question

Suppose we have a set $A$ of binary numbers with the same length $n$. For example (with $n=8$):

$A = \{ 10010011, 01011011, 00010010, 11110001\}$

Now, I want to find the binary number $z$ (also with $n$ bits) such that the minimum hamming distance between $z$ and any of these numbers is maximized.

i.e. $z = \arg \max \{ d(x,A) \mid x \in X\}$

with

$d(x,A) := \min \{ d(x,a) \mid a \in A\}$

where $X$ is the set of all $n$-bit binary numbers, and $d(x,a)$ the hamming distance between $x$ and $a$.

Is there any efficient algorithm for this?

Obviously, if $n$ is small enough we can brute-force this, but I'm looking for something more efficient when $n$ is large. The size of $A$, on the other hand, can be assumed relatively small, so any algorithm polynomial in $|A|$ is fine. Also, it does not have to be an exact algorithm. Any algorithm that approximates the correct answer is welcome too.