Diameter preserving surjection on alternate matrices

Let F be a field with |F| . 3, be the set of all m.m (m . 4) alternate matrices over F. The arithmetic distance of A,B . is d(A,B):= rank(A.B). If d(A,B) = 2, then A and B are said to be adjacent. The diameter of is max{d(A,B): A, B . }. Assume that .: s a map. We prove the following are equivalent: (a) . is a diameter preserving surjection in both directions, (b) . is both an adjacency preserving surjection and a diameter preserving map, (c) . is a bijective map which preserves the arithmetic distance