Binary Non-tiles

A subset V ⊆ Fn2 is a tile if F n 2 can be covered by disjoint translates of V . In other words, V is a tile if and only if there is a subset A ⊆ Fn2 such that V + A = Fn2 uniquely (i.e., v + a = v ′ + a′ implies that v = v′ and a = a′ where v, v′ ∈ V and a, a′ ∈ 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 binpacking 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 the context of hashing to find close matches, to show that none of them are, in fact, tiles.