On the binary codes with parameters of doubly-shortened 1-perfect codes

We show that any binary $(n=2^m-3, 2^{n-m}, 3)$ code $C_1$ is a part of an equitable partition (perfect coloring) $\{C_1,C_2,C_3,C_4\}$ of the $n$-cube with the parameters $((0,1,n-1,0)(1,0,n-1,0)(1,1,n-4,2)(0,0,n-1,1))$. Now the possibility to lengthen the code $C_1$ to a 1-perfect code of length $n+2$ is equivalent to the possibility to split the part $C_4$ into two distance-3 codes or, equivalently, to the biparticity of the graph of distances 1 and 2 of $C_4$. In any case, $C_1$ is uniquely embeddable in a twofold 1-perfect code of length $n+2$ with some structural restrictions, where by a twofold 1-perfect code we mean that any vertex of the space is within radius 1 from exactly two codewords.