On the admissible families of components of Hamming codes

TL;DR

This paper investigates the properties of Hamming code components, proposes methods to construct admissible families, and demonstrates how certain codes can be embedded into 1-perfect codes, expanding understanding of code structure and embedding possibilities.

Contribution

It introduces new constructions for admissible families of Hamming code components and proves embedding theorems for specific classes of codes into 1-perfect codes.

Findings

Every q-ary code of length m with minimum distance 5 can be embedded in a q-ary 1-perfect code.

Binary codes of length m + k with minimum distance 3k + 3 can be embedded in binary 1-perfect codes.

Properties and constructions of the i-components of Hamming codes are characterized.

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