How many binary sequences

You are given a set $B$ of binary sequences of length $K$. Every element of $B$ is a sequence of $K$ values, each of them $0$ or $1$.

An integer sequence $Z_i$ is built by this process.

1. Pick a sequence $X = (x_1, x_2, \dots, x_K)$ from $B$.
2. Pick a sequence $Y = (y_1, y_2, \dots, y_K)$ from $B$ with $\mathrm{dist}(X, Y) \le 2$. Here $\mathrm{dist}(X, Y)$ is the Hamming distance of the two sequences, the number of positions where the two values differ. For example, $\mathrm{dist}((1,0,1,1), (1,1,1,1)) = 1$ and $\mathrm{dist}((1,0,1,1,1,0,1), (1,0,0,1,0,0,1)) = 2$. You may pick the same element as both $X$ and $Y$.
3. Set $Z_i = (x_1 + y_1, x_2 + y_2, \dots, x_K + y_K)$.

For example, $Z_i = (1,0,1,2,2)$ can be built from $X = (1,0,0,1,1)$ and $Y = (0,0,1,1,1)$.

You are given $N$ integer sequences $Z_1, Z_2, \dots, Z_N$ built this way. Among all sets $B$ that can build all $N$ of them, find one with the fewest elements and print how many elements it has.

The first line contains $K$ and $N$, separated by one space. ($1 \le K \le 20$, $1 \le N \le 24$)

Each of the next $N$ lines contains one sequence $Z_i$. The $j$-th character of the $i$-th line is the value of $Z_{i,j}$, and there is no separator between characters. Each value is $0$, $1$, or $2$, and each line contains at most two $1$s, so every given $Z_i$ can be built by the process above.

Print the minimum number of elements of the set $B$.