# Obtaining binary perfect codes out of tilings

## TL;DR

This paper investigates conditions under which tilings of the Hamming cube produce perfect codes under specific metrics, characterizes these tilings, and introduces a method to construct new perfect codes through concatenation.

## Contribution

It identifies which known tilings lead to perfect codes with TS-metrics and proposes a new concatenation method for constructing perfect codes.

## Key findings

- Certain tilings of the Hamming cube yield perfect codes with TS-metrics.
- All TS-metrics turning known tilings into perfect codes are characterized.
- A new concatenation approach for building larger perfect codes is introduced.

## Abstract

A tiling of the $n$-dimensional Hamming cube 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 the $n$-dimensional Hamming cube which are determined by a weight which respects support of vectors (TS-metrics). We consider the known tilings of the Hamming cube and first determine which of them give rise to a perfect code. In the sequence, for those tilings that satisfy this condition, we determine all the TS-metrics that turns it into a perfect code. We also propose the construction of new perfect codes obtained by the concatenation of two smaller ones.