Sum of Dot Products

Let $a_1, \ldots, a_{16}$ be the list of $2^4 = 16$ distinct vectors which have 4 coordinates, whose values are either 0 or 1. What is the maximum possible value of $a_1 \cdot a_2 + a_3 \cdot a_4 + \cdots + a_{15} \cdot a_{16}$?

Details and assumptions

$u \cdot v$ represents the dot product of vectors.

Examples of vectors which have 4 coordinates and whose entries are either 0 or 1 are: $(0, 0, 0, 0), (1, 1, 1, 1), (0, 1, 0, 1), (1, 0, 0, 0)$.

The list is a set of all the 16 distinct vectors which satisfy the condition.