The existence of orthogonal resolutions of lines in AG(n, q)

Abstract A resolution (or packing) of the lines in AG(n, q) is a partitioning of the lines into classes R1, R2,…, Rt, t = (q n−1 ) (q−1) , such that each point of the geometry is on precisely one line of each class. One obvious resolution of AG(n, q) is the natural parallelism. In this paper, we are concerned with finding two resolutions R and R′ with the property that |R∩S| ⩽ 1 for all R ϵ R, S ϵ R′. The solutions R and R′ are said to be orthogonal. It is shown that a pair of orthogonal resolutions exists in AG(n, q) for n ⩾ 3 and all q a prime or prime power.