Abstract
A tiling of $\mathbb {F}_{2}^{n}$ (considered as a graph) gives rise to a perfect code (according to a given metric) if the basic tile is a metric ball. We are concerned with metrics on $\mathbb {F}_{2}^{n}$ that are determined by a weight which respects the support of vectors (TS-metrics) and in this case, a perfect code is called TS-perfect. We consider all the tilings of $\mathbb {F}_{2}^{n}$ by tiles with up to 8 elements (which are classified in the literature) and determine which of them give rise to a TS-perfect code. In the sequence, for those tilings that give rise to a perfect code for some TS-metric, we classify the TS-metrics (up to equivalence) that turn it into a perfect code. Also, we give a general construction to obtain TS-perfect codes out of given TS-perfect codes, by proving that concatenation of two TS-perfect codes is by itself a TS-perfect code.
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