Equivalence classes of small tilings of the Hamming cube
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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 F_2^n, the small tilings, that is, tilings considering tiles with 8=2^3 elements, were classified in 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.
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