Perfect binary codes of infinite length

A subset C of the infinite-dimensional Boolean cube {0,1} 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} . A perfect binary code in the zero layer {0,1}^, consisting of all vectors of the cube {0,1} 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}^ 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} is hypercontinuum.

Keywords

Authors

References