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.

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