Geodesic
Perfect binary codes of infinite length with complete system of triples
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 877-888.

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An infinite-dimensional binary cube $\{0,1\}_0^{\mathbb N}$ consists of all sequences $u = (u_1,u_2,\dots)$, where $u_i= 0,1$, and all $u_i =0$ except some finite set of indices $i \in \mathbb N$. A subset $C \subset \{0,1\}_0^{\mathbb N}$ is called a perfect binary code with distance 3 if all balls of radius 1 (in the Hamming metric) with centers in $C$ are pairwise disjoint and their union covers this binary cube. We say that the perfect code $C$ has the complete system of triples if $C + C$ contains all vectors of $\{0,1\}_0^{\mathbb N}$ having weight 3. In this article we construct perfect binary codes having the complete system of triples (in particular, such codes are nonsystematic). These codes can be obtained from the Hamming code $H^\infty$ by switchings a some family of disjoint components ${\mathcal B} = \{R_1^{u_1},R_2^{u_2},\dots\}$. Unlike the codes of finite length, the family $\mathcal B$ must obey the rigid condition of sparsity. It is shown particularly that if the family of components $\mathcal B$ does not satisfy the condition of sparsity then it can generate a perfect code having non-complete system of triples.
Keywords: perfect binary code, component, complete system of triples, nonsystematic code, condition of sparsity. 
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     author = {S. A. Malyugin},
     title = {Perfect binary codes of infinite length with complete system of triples},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {877--888},
     publisher = {mathdoc},
     volume = {14},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2017_14_a75/}
}
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S. A. Malyugin. Perfect binary codes of infinite length with complete system of triples. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 877-888. http://geodesic.mathdoc.fr/item/SEMR_2017_14_a75/

[1] Avgustinovich S. V., Solov'eva F. I., “On the nonsystematic perfect binary codes”, Problems Inform. Transmission, 32:3 (1996), 258–261 | MR | Zbl

[2] Vasil'ev Yu. L., “On Nongroup Close-Packed Codes”, Problems of Cybernetics, 8 (1962), 337–339 | MR | Zbl

[3] Malyugin S. A., “On enumeration of the perfect binary codes of length 15”, Discrete Applied Mathematics, 135:1–3 (2004), 161–181 | DOI | MR

[4] Malyugin S. A., “Perfect binary codes of infinite length”, J. Appl. Indust. Math., 8:4 (2017), 552–556 | DOI | MR

[5] Potapov V. N., “Infinite-Dimensional Quasigroups of finite orders”, Math. Notes, 93:3 (2013), 479–486 | DOI | MR | Zbl

[6] Romanov A. M., “On the construction of perfect nonlinear binary codes by symbol inversion”, Discretn. Anal. Issled. Oper. Ser. 1, 4:1 (1997), 46–52 (in Russian) | MR | Zbl

[7] Phelps K. T., LeVan M. J., “Kernels of nonlinear Hamming codes”, Designs, Codes and Cryptogr., 6:3 (1995), 247–257 | DOI | MR | Zbl

[8] Phelps K. T., LeVan M. J., “Nonsystematic perfect codes”, SIAM J. Discrete Math., 12:1 (1999), 27–34 | DOI | MR | Zbl