This article is cited in 3 papers

On the admissible families of components of Hamming codes

A. M. Romanov

S. L. Sobolev Institute of Mathematics, SB RAS, Novosibirsk, Russia

Abstract: We describe the properties of the $i$-components of Hamming codes. We suggest constructions of the admissible families of components of Hamming codes. It is shown that every $q$-ary code of length $m$ and minimum distance 5 (for $q=3$ the minimum distance is 3) can be embedded in a $q$-ary 1-perfect code of length $n=(q^m-1)/(q-1)$. It is also demonstrated that every binary code of length $m+k$ and minimum distance $3k+3$ can be embedded in a binary 1-perfect code of length $n=2^m-1$. Bibliogr. 5.

Keywords: Hamming code, 1-perfect code, $q$-ary code, binary code, $i$-component.

UDC: 519.176

Received: 13.05.2011
Revised: 21.11.2011

 English version: Journal of Applied and Industrial Mathematics, 2012, 6:3, 355–359

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