Equivalence classes of small tilings of the Hamming cube

TL;DR

This paper classifies small tilings of the Hamming cube under coordinate permutations, reducing 193 known tiles to 15 equivalence classes using explicit permutation mappings.

Contribution

It introduces a classification of small tilings of the Hamming cube under coordinate permutation isometries, simplifying the understanding of their equivalence classes.

Findings

193 tiles reduced to 15 classes under permutations

Explicit permutations identified for each class

Simplifies the classification of small tilings

Abstract

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.