On lifting perfect codes

TL;DR

This paper studies the properties of lifted perfect codes, showing they are completely regular and characterizing Hamming codes as the unique codes that maintain this property after lifting the ground field.

Contribution

It introduces a new class of completely regular codes obtained by lifting perfect codes and proves the uniqueness of Hamming codes in this context.

Findings

Lifted perfect codes are completely regular with specific covering radius.

Intersection numbers of these codes are explicitly computed.

Hamming codes are uniquely characterized as the only codes that remain completely regular after lifting.

Abstract

In this paper we consider completely regular codes, obtained from perfect (Hamming) codes by lifting the ground field. More exactly, for a given perfect code C of length n=(q^m-1)/(q-1) over F_q with a parity check matrix H_m, we define a new code C_{(m,r)} of length n over F_{q^r}, r > 1, with this parity check matrix H_m. The resulting code C_{(m,r)} is completely regular with covering radius R = min{r,m}. We compute the intersection numbers of such codes and, finally, we prove that Hamming codes are the only codes that, after lifting the ground field, result in completely regular codes.