A new type of coding problem

Let X be an n-element finite set, and 0 < k < n/2 an integer. Suppose that {A1, B1} and {A(2), B-2} are pairs of disjoint k-element subsets of X (that is, \A(1)\ = \B-1\= \A(2)\ = \B-2\ = k, A(1) boolean AND B-1 = phi, A(2) boolean AND B-2 = circle divide). Define the distance between these pairs by d({A(1), B-1}, {A(2), B-2}) = min{\A(1) – A(2) \ + \B-1 – B-2\ + \A(1) – B-2\ + \B-1 – A(2)\}. It is known ([2]) 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 between any two pairs is at least k. Here we answer questions arising for distances larger than k.