Multiple Packing in Sum-type Metric Spaces

We present the exact solution of the asymptotics of the multiple packing problem in a finite space with a sum-type metric. 1 Consider the space Q of n−tuples over a finite set Q = [q] = {1, 2, . . . , q} (by standard convention in Combinatorics) and a sum-type metric d(x, y) = ∑n i=1 d(xi, yi), x = (x1, . . . , xn), y = (y1, . . . , xn). Let Bn(x, r) := {y ∈ Q : d(x, y) ≤ r} be the ball of radius r in Q with the center in y. We say that a subset An ⊂ Q is an L−packing by the balls of radius r if max x∈Qn ∣