On linear completely regular codes with covering radius $\rho=1$. Construction and classification

TL;DR

This paper classifies and constructs linear completely regular codes with covering radius 1, focusing on cases with minimum distance 1 or 2, and proves their complete transitivity.

Contribution

It provides a complete characterization and construction of linear completely regular codes with radius 1 for minimum distances 1 and 2, extending known results beyond perfect codes.

Findings

Codes with d=3 are perfect and well-known.

All studied codes are shown to be completely transitive.

The paper characterizes linear codes with d=1 or 2 and radius 1.

Abstract

Completely regular codes with covering radius $\rho=1$ must have minimum distance $d\leq 3$. For $d=3$, such codes are perfect and their parameters are well known. In this paper, the cases $d=1$ and $d=2$ are studied and completely characterized when the codes are linear. Moreover, it is proven that all these codes are completely transitive.