Prikladnaya Diskretnaya Matematika. Supplement
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Prikl. Diskr. Mat. Suppl., 2015, Issue 8, Pages 117–120 (Mi pdma227)  

This article is cited in 1 scientific paper (total in 1 paper)

Applied Theory of Coding, Automata and Graphs

Perfect binary codes of infinite length

S. A. Malyugin

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

Abstract: A subset $C$ of the infinite-dimensional Boolean cube $\{0,1\}^\mathbb N$ is called a perfect binary code with distance 3 if all balls of radius 1 (in the Hamming metric) with centres in $C$ are pairwise disjoint and their union covers the cube $\{0,1\}^\mathbb N$. A perfect binary code in the zero layer $\{0,1\}^\mathbb N_0$, consisting of all vectors of the cube $\{0,1\}^\mathbb N$ having finite supports, is defined similarly. It is proved that the cardinality of the set of all equivalence classes of perfect binary codes in the zero layer $\{0,1\}^\mathbb N_0$ is continuum. At the same time, the cardinality of the set of all equivalence classes of perfect binary codes in the whole cube $\{0,1\}^\mathbb N$ is hypercontinuum.

Keywords: perfect binary codes, Hamming code, Hamming distance, Vasil'ev codes, equivalence classes, continuum, hypercontinuum.

DOI: https://doi.org/10.17223/2226308X/8/45

Full text: PDF file (619 kB)
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UDC: 519.7

Citation: S. A. Malyugin, “Perfect binary codes of infinite length”, Prikl. Diskr. Mat. Suppl., 2015, no. 8, 117–120

Citation in format AMSBIB
\Bibitem{Mal15}
\by S.~A.~Malyugin
\paper Perfect binary codes of infinite length
\jour Prikl. Diskr. Mat. Suppl.
\yr 2015
\issue 8
\pages 117--120
\mathnet{http://mi.mathnet.ru/pdma227}
\crossref{https://doi.org/10.17223/2226308X/8/45}


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    This publication is cited in the following articles:
    1. S. A. Malyugin, “Perfect binary codes of infinite length”, J. Appl. Industr. Math., 11:2 (2017), 227–235  mathnet  crossref  crossref  elib
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