Perfect binary codes of infinite length with complete system of triples

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.