Minimal generating set size

[Dan Dima]

Take all the points having integer coordinates (i,j,k) , 0 <= i,j,k <= N -
inside a cube C of length N.

Determine a(N) as the size of a minimal generating set such that:
Find a minimal generating set A of size a(N) such that any point in C lies
on some line generated by two points from A.
I have seen lower & upper bounds delivered for the 2D-case.
Does anyone know if there is a general formula?

Best regards,
Dan
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