Binary Non-tiles
Discrete Mathematics
2011-08-02 v2 Information Theory
Combinatorics
math.IT
Abstract
A subset V of GF(2)^n is a tile if GF(2)^n can be covered by disjoint translates of V. In other words, V is a tile if and only if there is a subset A of GF(2)^n such that V+A = GF(2)^n uniquely (i.e., v + a = v' + a' implies that v=v' and a=a' where v,v' in V and a,a' in A). In some problems in coding theory and hashing we are given a putative tile V, and wish to know whether or not it is a tile. In this paper we give two computational criteria for certifying that V is not a tile. The first involves impossibility of a bin-packing problem, and the second involves infeasibility of a linear program. We apply both criteria to a list of putative tiles given by Gordon, Miller, and Ostapenko in that none of them are, in fact, tiles.
Cite
@article{arxiv.0911.1388,
title = {Binary Non-tiles},
author = {Don Coppersmith and Victor S. Miller},
journal= {arXiv preprint arXiv:0911.1388},
year = {2011}
}