On the structure of cube tiling codes

Abstract

Let S be a set of arbitrary objects, and let Sd={v1...vd:vi∈S}. A polybox code is a set V⊂Sd with the property that for every two words v,w∈V there is i∈[d] with vi′=wi, where a permutation s↦s′ of S is such that s′′=(s′)′=s and s′≠s. If |V|=2d, then V is called a cube tiling code. Cube tiling codes determine 2-periodic cube tilings of Rd or, equivalently, tilings of the flat torus Td={(x1,…,xd)(mod2):(x1,…,xd)∈Rd} by translates of the unit cube as well as r-perfect codes in Z4r+2d in the maximum metric. By a structural result, cube tiling codes for d=4 are enumerated. It is computed that there are 27,385 non-isomorphic cube tiling codes in dimension four, and the total number of such codes is equal to 17,794,836,080,455,680. Moreover, some procedure of passing from a cube tiling code to a cube tiling code in dimensions d≤5 is given.