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:
There is a set A of size a(N) such that any point in C lies on some line
generated by two points from A.
Any other set of size smaller than a(N) does not have this property.
I have seen some lower & upper bounds delivered for the 2D-case.
Does anyone know if there is a general formula (for both 2D, 3D or even
k-dim.space)?

Best regards,
Dan
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