# Metrics which turn tilings into binary perfect codes

## TL;DR

This paper investigates specific metrics on the Hamming cube that transform certain tilings into perfect codes, characterizing these metrics and providing methods to generate new perfect codes based on them.

## Contribution

It characterizes all TS-metrics that turn particular tilings into perfect codes and introduces procedures to construct new perfect codes from existing ones.

## Key findings

- Identifies which tilings can be perfect codes under TS-metrics
- Provides a complete characterization of such metrics
- Proposes methods to derive new perfect codes from known ones

## Abstract

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.