Using error-correcting codes to construct solvable pebbling distributions

Abstract

In pebbling problems, pebbles are placed on the vertices of a graph. A pebbling move consists of removing two pebbles from one vertex, throwing one away, and moving the other pebble to an adjacent vertex. We say a distribution D is solvable if starting from D, we can move a pebble to any vertex by a sequence of pebbling moves. The optimal pebbling number of a graph G is the smallest number of pebbles in a solvable distribution on G.It is known that every solvable distribution on the n-dimensional hypercube Qn contains at least (43)n pebbles. Fu, Huang, and Shiue, building on the work of Moews, used probabilistic methods to show that there are solvable distributions where the number of pebbles is in O((43)nn32), but hitherto, the number of pebbles in the best constructed distributions was in O(1.377n).We use error-correcting codes to construct solvable distributions of pebbles on Qn in which the number of pebbles is in O(1.34n).