Cardinality of a set in a field

Question

If $$A = \bigg\{\sum_{i=0}^4a_i+b_i \ \big| \sum_{i=0}^4a_i+b_i = 0: a_i, b_i \in \mathbb{F}_{2^k}\bigg\}$$ where $\mathbb{F}_{2^k}$ is a field with $2^k$ elements for some positive integer $k$. I want to prove that cardinality of $A$ is $2^{9k}$. How to prove?