Projective space $PG(3,2)$ has nice 5-tuples of points like $(1,0,0,0)$, $(0,1,0,0)$, $(0,0,1,0)$, $(0,0,0,1)$, $(1,1,1,1).$ This 5-tuple is "nice" because these points with their pairwise sums cover whole space $PG(3,2)$.
What is the name for such 5-tuples? In what combinatorial problems do they arise?
On
This set of points is an example of a frame. More precisely, a frame is a set of $n+2$ points in a $n$-dimensional projective space such that no hyperplane contains $n+1$ of them.
Frames are nice in general because we can describe a matrix for a linear transformation by considering where it sends the points in a frame (whereas in most situations a set of points forming a basis won't allow this).
The property of the pairwise sums giving the whole space is, I think, not an especially interesting property, and I don't think there is a nice generalization over fields with order greater than 2.
This works for any basis of $V(4,2)$. Let $v_1$, $v_2$, $v_3$, and $v_4$ be any such basis, and let $v_5=v_1+v_2+v_3+v_4$. Then the other ten points are the spans of the vectors: $v_1+v_2$, $v_1+v_3$, $v_1+v_4$, $v_2+v_3$, $v_2+v_4$ and $v_3+v_4$, as well as $v_1+v_5 = v_2+v_3+v_4$, $v_2+v_5$, $v_3+v_5$ and $v_4+v_5$.