The existence of orthogonal resolutions of lines in AG(n,q) (Q1096174)

In a finite affine geometry, AG(n,q), q being a prime-power, there is always a resolution, i.e. a partition of the set of lines into classes (the spreads) such that each point is on precisely one line of each class. The natural parallelism induces a resolution. But when the dimension, n, is superior to 2, there are other resolutions. The authors call two resolutions, R and S, orthogonal if for each \(X\in R\) and each \(Y\in S\) \(card (X\cap Y)\leq 1.\) They had studied various particular cases in their previous papers [Congr. Numerantium 28, 399-407 (1980; Zbl 0456.05020); Aequationes Math. 23, 242-251 (1981; Zbl 0511.05023); Discrete Math. 41, 17-28 (1982; Zbl 0509.05013); Ann. Discrete Math. 18, 355-375 (1983; Zbl 0504.05014)]. In the present paper they eventually prove the existence of at least one pair of orthogonal resolutions in each AG(n,q) for \(n>2\).