Let $c_0, \cdots, c_{n-1}$ be elements of $\mathbb{F_{2^k}}$, find the sum $\sum\limits_{0\leq i < j <n}c_ic_j$.
It is true that if $k=1$ and $d$ be the number of non-zero elements, then $$\sum\limits_{0\leq i < j <n}c_ic_j=\bigg(\begin{matrix}d\\2\end{matrix}\bigg) \mod 2$$
What happens if $k>1$.