# On completely regular and completely transitive codes derived from Hamming codes

## TL;DR

This paper explores how partitioning the parity-check matrix of a Hamming code leads to new classes of completely regular and transitive codes, expanding the understanding of their structure and properties.

## Contribution

It introduces a novel method of constructing completely regular and transitive codes from Hamming codes via matrix partitioning, establishing their properties and infinite families.

## Key findings

- Supplementary codes of Hamming codes are completely regular and transitive.
- If one code is completely regular with radius 2, the other shares this property.
- Constructs infinite families of quasi-perfect uniformly packed codes.

## Abstract

Given a parity-check matrix $H_m$ of a $q$-ary Hamming code, we consider a partition of the columns into two subsets. Then, we consider the two codes that have these submatrices as parity-check matrices. We say that anyone of these two codes is the supplementary code of the other one. We obtain that if one of these codes is a Hamming code, then the supplementary code is completely regular and completely transitive. If one of the codes is completely regular with covering radius $2$, then the supplementary code is also completely regular with covering radius at most $2$. Moreover, in this case, either both codes are completely transitive, or both are not. With this technique, we obtain infinite families of completely regular and completely transitive codes which are quasi-perfect uniformly packed.