Binary Non-tiles

TL;DR

This paper introduces two computational methods to determine whether a subset of GF(2)^n is a tile, and applies them to show that certain candidate tiles are not actual tiles, advancing understanding in coding theory and hashing.

Contribution

It provides novel computational criteria using bin-packing and linear programming to certify non-tileness of subsets in GF(2)^n, with practical applications.

Findings

Both criteria successfully certify that all tested putative tiles are not tiles.

The methods are effective tools for analyzing tiling problems in coding theory.

Application to known candidates confirms their non-tiling nature.

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.