Geodesic
On the admissible families of components of Hamming codes
Diskretnyj analiz i issledovanie operacij, Tome 19 (2012) no. 2, pp. 84-91.

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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. 
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A. M. Romanov. On the admissible families of components of Hamming codes. Diskretnyj analiz i issledovanie operacij, Tome 19 (2012) no. 2, pp. 84-91. http://geodesic.mathdoc.fr/item/DA_2012_19_2_a5/

[1] Romanov A. M., “O razbieniyakh $q$-ichnykh kodov Khemminga na neperesekayuschiesya komponenty”, Diskret. analiz i issled. operatsii. Ser. 1, 11:3 (2004), 80–87 | MR | Zbl

[2] Avgustinovich S. V., Krotov D. S., “Embedding in a perfect code”, J. Comb. Des., 17:5 (2009), 419–423 | DOI | MR | Zbl

[3] Etzion T., Vardy A., “Perfect binary codes: Constructions, properties and enumeration”, IEEE Trans. Inf. Theory, 40:3 (1994), 754–763 | DOI | MR | Zbl

[4] Phelps K. T., Villanueva M., “Ranks of $q$-ary 1-perfect codes”, Des. Codes Cryptography, 27:1–2 (2002), 139–144 | DOI | MR | Zbl

[5] Romanov A. M., “Survey of methods for construction of nonlinear perfect binary codes”, J. Appl. Industr. Math., 2:2 (2008), 252–269 | DOI