# On completely regular codes of covering radius 1 in the halved hypercubes

## TL;DR

This paper explores the construction of completely regular codes with covering radius 1 in halved hypercubes, focusing on equitable partitions and perfect colorings, contributing to the understanding of their combinatorial structures.

## Contribution

It provides new constructions and characterizations of completely regular codes of covering radius 1 in halved hypercubes, linking them to equitable 2-partitions and perfect colorings.

## Key findings

- New constructions of completely regular codes in halved hypercubes
- Characterization of equitable 2-partitions related to these codes
- Insights into the structure of perfect colorings in halved hypercubes

## Abstract

We consider constructions of covering-radius-1 completely regular codes, or, equivalently, equitable 2-partitions (regular 2-partitions, perfect 2-colorings), of halved n-cubes. Keywords: completely regular code, equitable partition, regular partition, partition design, perfect coloring, halved hypercube.