How to find a 2-wise independent hash family that is not 3-wise independent? I'm trying to find a family of hash functions mapping $\{1, 2, ..., 2^n\}$ to $\{0, 1\}$ that is 2-wise independent but not 3-wise independent. Any ideas on that?

I know two 2-wise independent families but I think they are 3-wise independent as well:

1) Let x be a vector in $\{0, 1\}^n$. The hash functions are $h_{a, b}(x)=a^\top x+b \mod  2$ for all $a\in\{0, 1\}^n$ and $b\in\{0, 1\}$.

2) Consider $GF(2^n)$. The hash functions are $$h_{a, b}(x)= \text{the last bit of }a \cdot x+b $$ for all $a, b\in GF(2^n)$.
The smallest example I could find is
$$
0001 \\
0010 \\
0100 \\
1000 \\
0011 \\
0101 \\
0110 \\
1001 \\
1010 \\
1100 \\
1111 \\
1111
$$
I'll leave you to generalize this to larger $n$.