On the admissible families of components of Hamming codes

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 = (qm−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 = 2m − 1.