Perfect binary codes of infinite length

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.