On lifting perfect codes
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=(qm-1)/(q-1) over Fq with a parity check matrix Hm, we define a new code C(m,r) of length n over Fqr, r > 1, with this parity check matrix Hm. The resulting code C(m,r) is completely regular with covering radius R = minr,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.
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