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This article is cited in 2 scientific papers (total in 2 papers)
Discrete mathematics and mathematical cybernetics
Perfect binary codes of infinite length with complete system of triples
S. A. Malyugin Sobolev Institute of Mathematics,
pr. Koptyuga, 4,
630090, Novosibirsk, Russia
Abstract:
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.
Received July 26, 2017, published September 14, 2017
Citation:
S. A. Malyugin, “Perfect binary codes of infinite length with complete system of triples”, Sib. Èlektron. Mat. Izv., 14 (2017), 877–888
Linking options:
https://www.mathnet.ru/eng/semr831 https://www.mathnet.ru/eng/semr/v14/p877
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Abstract page: | 135 | Full-text PDF : | 27 | References: | 35 |
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