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 Qn 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 Qn : d(x, y) r} be the ball of radius r in Qn with the center in y. We say that a subset An Qn is an L-packing by the balls of radius r if max xQn Bn(x, r) An L or equivalently for an arbitrary set of L + 1 n-tuples {u1, . . . , uL+1} An L+1 j=1 Bn(uj, r) = . We refer to papers [1-6] as literature about different properties and asymptotics of L-packings. L-packing finds applications in coding theory. Adopting the terminology from there we can consider the model when n-tuples from the set An are transmitted over the channel where up to t errors can occur (up to t coordinates of the output y Qn of the channel differ from the corr...

• Type: Journal
• Year: 2008
• Where: DAM
• Authors: Rudolf Ahlswede, Vladimir Blinovsky
• Keywords: DAM 2008, Multiple Packing Problem, Space Qn, Subset An Qn