Geodesic
Perfect multiple coverings of hypercube
Diskretnyj analiz i issledovanie operacij, Tome 19 (2012) no. 4, pp. 60-65.

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A subset $C$ of graph vertices is called a perfect $k$-multiple covering with a radius $r$ if every vertex of this graph is within distance $r$ from exactly $k$ vertices of $C$. We give a criterion based on parameters of a perfect coloring. This criterion determines whether the perfect coloring is a perfect multiple covering with fixed radius $r\geq1$ of some multiplicity. Bibliogr. 13.
Mots-clés : hypercube
Keywords: perfect coloring, perfect code, perfect multiple coverings. 
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K. V. Vorob'ev. Perfect multiple coverings of hypercube. Diskretnyj analiz i issledovanie operacij, Tome 19 (2012) no. 4, pp. 60-65. http://geodesic.mathdoc.fr/item/DA_2012_19_4_a4/

[1] Vorobev K. V., Fon-Der-Flaass D. G., “O sovershennykh 2-raskraskakh giperkuba”, Sib. elektron. mat. izv., 7 (2010), 67–75 | MR

[2] Zinovev V. A., Leontev V. K., “O sovershennykh kodakh”, Probl. peredachi inform., 8:1 (1972), 26–35 | MR | Zbl

[3] Krotov D. S., “O sovershennykh raskraskakh polovinnogo 24-kuba”, Diskret. analiz i issled. operatsii, 15:5 (2008), 35–46 | MR | Zbl

[4] Potapov V. N., “O sovershennykh raskraskakh buleva $n$-kuba i korrelyatsionno-immunnykh funktsiyakh maloi plotnosti”, Sib. elektron. mat. izv., 7 (2010), 372–382 | MR

[5] Fon-Der-Flaass D. G., “Sovershennye 2-raskraski 12-mernogo kuba, dostigayuschie granitsy korrelyatsionnoi immunnosti”, Sib. elektron. mat. izv., 4 (2007), 292–295 | MR | Zbl

[6] Fon-Der-Flaass D. G., “Sovershennye 2-raskraski giperkuba”, Sib. mat. zhurn., 48:4 (2007), 923–930 | MR | Zbl

[7] Cohen G. D., Honkala I. S., Litsyn S. N., Lobstein A. C., Covering codes, Elsevier, Amsterdam, 1997, 542 pp. | MR

[8] Fon-Der-Flaass D. G., “A bound on correlation immunity”, Sib. Electron. Math. Reports, 4 (2007), 133–135 | MR | Zbl

[9] Habsieger L., “Integer zeros of q-Krawtchouk polynomials in classical combinatorics”, J. Adv. Appl. Math., 27:2–3 (2001), 427–437 | DOI | MR | Zbl

[10] Krasikov I., Litsyn S., “On integral zeros of Krawtchouk polynomials”, J. Comb. Theory Ser. A, 74:1 (1996), 71–99 | DOI | MR | Zbl

[11] Nagell T., “The Diophantine equation $x^2+7=2^n$”, Ark. Mat., 4:2–3 (1960), 185–187 | MR

[12] Tietavainen A., “On the nonexistence of perfect codes over the finite fields”, SIAM J. Appl. Math., 24 (1973), 88–96 | DOI | MR

[13] Van Wee G. J. M., Cohen G. D., Litsyn S. N., “A note on perfect multiple coverings of the Hamming space”, IEEE Trans. Inf. Theory, 37:3 (1991), 678–682 | DOI | Zbl