Prove that this family of hash function is $3$-wise independent, but not $4$-wise independent
Hint for refuting 4-wise independence: Suppose for simplicity that $c = 2$, pick some $\alpha \neq \beta$ of length $r$, and consider the four inputs $(\alpha,\alpha),(\alpha,\beta),(\beta,\alpha),(\beta,\beta)$.

Hint for proving 3-wise independence: Let the three inputs be $x,y,z$. If there is some coordinate $i$ such that $x_i,y_i,z_i$ are all different then we are done. Otherwise, without loss of generality there are coordinates $i,j$ such that $x_i = y_i \neq z_i$ and $x_j = z_j \neq y_j$, and we are again done.