A tiling of F2n(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 F2nthat 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 F2nby 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.

In this work, we consider tilings of the Hamming cube and look for metrics which turn the tilings into a perfect code. We consider the family of metrics which are determined by a weight and are compatible with the support of vectors (TS-metrics). We determine which of the tilings with small tiles or high rank can be a perfect code for some TS-metric and we characterize all such metrics. Finally, we show some procedures to obtain new perfect codes (relatively to TS-metrics) out of existing ones.