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Journal of Applied and Industrial Mathematics, 2017, 11:2, 227-235
Volume 24, No 2, 2017, P. 53-67
UDC 519.8
S. A. Malyugin
Perfect binary codes of infinite length
Abstract:
A subset $C$ of infinite-dimensional binary cube 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 cover this binary cube. Similarly, we can define a perfect binary code in zero layer, consisting of all vectors of infinite-dimensional binary cube having finite supports. In this article we prove that the cardinality of all cosets of perfect binary codes in zero layer is the cardinality of the continuum. Moreover, the cardinality of all cosets of perfect binary codes in the whole binary cube is equal to the cardinality of the hypercontinuum.
Bibliogr. 9.
Keywords: perfect binary code, Hamming code, Vasil’ev code, component, continuum, hypercontinuum.
DOI: 10.17377/daio.2017.24.535
Serguey A. Malyugin 1
1. Sobolev Institute of Mathematics,
4 Koptyug Ave., 630090 Novosibirsk, Russia
e-mail: mal@math.nsc.ru
Received 31 March 2016
Revised 29 August 2016
References
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