On the admissible families of components of Hamming codes

Details On the admissible families of components of Hamming codes On the admissible families of components of Hamming codes

2012-02-02

arxiv.org/abs/1202.0349v1

Computer Science

Information Theory

7 pages

Scientific paper

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 $n = 2^{m}-1$.

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