A coding problem for pairs of subsets

Let X be an n–element finite set, 0 < k ≤ n/2 an integer. Suppose that {A1, A2} and {B1, B2} are pairs of disjoint k-element subsets of X (that is, |A1| = |B1| = |A2| = |B2| = k,A1 ∩ A2 = ∅, B1 ∩ B2 = ∅). Define the distance of these pairs by d({A1, A2}, {B1, B2}) = min{|A1 − B1| + |A2 − B2|, |A1 −B2|+ |A2 −B1|}. Let C(n, k, d) be the maximum size of a family of pairs of disjoint subsets, such that the distance of any two pairs is at least d. Here we establish a conjecture from [2] concerning the asymptotic formula for C(n, k, d) by using the randomized packing theorem of J. Kahn [10] for k, d are fixed and n → ∞. Also, an infinite number of exact results are given by using special difference sets of integers. Finally, the questions discussed above are put into a more general context and a number of coding theory type problems are proposed. 1 The transportation distance Let X be a finite set of n elements, if it is convenient we identify it with the set [n] := {1, 2, . . . , n}. The family of the k-sets of an underlying set X is denoted by ( X k ) . For 0 < k ≤ n/2 let Y be the family of unordered disjoint pairs {A1, A2} of k-element subsets of X (that is, |A1| = |A2| = k,A1 ∩ A2 = ∅). We define the transportation distance d on Y by d({A1, A2}, {B1, B2}) = min{|A1 −B1|+ |A2 −B2|, |A1 −B2|+ |A2 −B1|}. (1) In fact, this is an instance of a more general notion. Whenever (Z, ρ) is a metric space, we can define a metric ρ on Z by ρ((x1, . . . , xs), (y1, . . . , ys)) = min π∈Ss s ∑ i=1 ρ(xi, yπ(i)). (2) ∗Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB, UK, and Department of Mathematical Sciences, University of Memphis, Memphis TN 38152, USA. †Alfréd Rényi Institute of Mathematics, 13–15 Reáltanoda Street, 1053 Budapest, Hungary. E-mail: [email protected]. Research supported in part by the Hungarian National Science Foundation OTKA 104343, and by the European Research Council Advanced Investigators Grant 267195. ‡Computer Science Institute of Charles University, Malostranské nám. 25, 118 00 Praha 1. Czech Republic. E-mail: [email protected]. Research supported by GAČR grant number P201/12/P288 and partially done while this author visited the Rényi Institute. §Rényi Institute, Hungarian Academy of Sciences, Budapest, Reáltanoda u. 13–15, 1053 Hungary. E-mail: ohkatona@renyi,mta.hu. Research was supported by the Hungarian National Foundation OTKA T029255. This work was done while this author visited the University of Memphis. ¶Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB, UK. [email protected]. This work was done while this author visited the University of Memphis.