On lifting perfect codes

Josep Rifà, Victor A. Zinoviev

Research output: Contribution to journalArticleResearchpeer-review

8 Citations (Scopus)

Abstract

In this paper, we consider completely regular codes, obtained from perfect (Hamming) codes by lifting the ground field. More exactly, for a given Hamming code C of length n=(qm - 1)/(q - 1) over double-struk F sign q with a parity check matrix Hm, we define a new linear code C(m,r) of length n over double-struk F signqr, r ≥ 2, with this parity check matrix Hm. The resulting code C (m,r) is completely regular with covering radius ρ = min{r,m}. We compute the intersection numbers of such codes and we prove that Hamming codes are the only codes that, after lifting the ground field, result in completely regular codes. Finally, we also prove that extended perfect (Hamming) codes, for the case when extension increases their minimum distance, are the only codes that, after lifting the ground field, result in uniformly packed (in the wide sense) codes. © 2011 IEEE.
Original languageEnglish
Article number6006603
Pages (from-to)5918-5925
JournalIEEE Transactions on Information Theory
Volume57
DOIs
Publication statusPublished - 1 Sep 2011

Keywords

  • Completely regular codes
  • Hamming codes
  • covering radius
  • extended Hamming codes
  • intersection numbers
  • uniformly packed codes

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