Equivalence classes of small tilings of the Hamming cube

The study of tilings is a major problem in many mathematical instances, which is studied in two main different approaches: when considering the existence (or obstructions to the existence) of a tiling with a given tile and the other considering classification of tilings. Considering the Hamming cube $\mathbb{F}_2^n$, the small tilings, that is, tilings considering tiles with $8=2^3$ elements, were classified in \cite{vardy}. The authors list a total of $193$ different tiles. As the authors noted, many of those tiles can be obtained one from the other by a linear map. In this work, we are concerned with a particular class of linear maps, the class of permutations of coordinates. This is of interest since a permutation is an isometry of the Hamming cube, considering the Hamming metric. We show here that, up to an isometry, all those $193$ tiles can be reduced to $15$ classes. The proof is done by explicitly showing the permutation (represented in cycles) that identify each tile with a given representative.