A new type of coding problem

Let X be an n–element finite set, 0 < k < n/2 an integer. Suppose that {A1, B1} and {A2, B2} are pairs of disjoint k-element subsets of X (that is, |A1| = |B1| = |A2| = |B2| = k,A1 ∩ B1 = ∅, A2 ∩ B2 = ∅). Define the distance of these pairs by d({A1, B1}, {A2, B2}) = min{|A1−A2|+ |B1−B2|, |A1−B2|+ |B1−A2|}. It is known ([2]) that this is really a distance on the space of such pairs and that the family of all k–element subsets of X can be paired (with one exception if their number is odd) in such a way that the distance of the pairs is at least k. Here we answer questions arising for distances larger than k.