Obtaining Binary Perfect Codes Out of Tilings

A tiling of 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 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 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.