On lifting perfect codes

Understand

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.