On the structure of cube tiling codes

Abstract: Let $S$ be a set of arbitrary objects, and let $S^d={v_1...v_d\colon v_i\in S}$. A polybox code is a set $V\subset S^d$ with the property that for every two words $v,w\in V$ there is $i\in [d]$ with $v_i'=w_i$, where a permutation $s\mapsto s'$ of $S$ is such that $s''=(s')'=s$ and $s'\neq s$. If $|V|=2^d$, then $V$ is called a cube tiling code. Cube tiling codes determine $2$-periodic cube tilings of $\mathbb{R}^d$ or, equivalently, tilings of the flat torus $\mathbb{T}^d={(x_1,\ldots ,x_d)({\rm mod} 2):(x_1,\ldots ,x_d)\in \mathbb{R}^d}$ by translates of the unit cube as well as $r$-perfect codes in $\mathbb{Z}^d_{4r+2}$ 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 $d\leq 5$ is given.