On the admissible families of components of Hamming codes

Abstract

In this paper, 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 shown that every binary code of length $m+k$ and minimum distance $3k+3$ can be embedded in a binary 1-perfect code of length $n=2^m-1$.

arXiv comment7 pages