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

We show that any binary (n = 2k − 3, 2n−k, 3) code C1 is a cell of an equitable partition (perfect coloring) (C1, C2, C3, C4) of the n-cube with the quotient matrix ((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 C1 to a 1-perfect code of length n + 2 is equivalent to the possibility to split the cell C4 into two distance-3 codes or, equivalently, to the biparticity of the graph of distances 1 and 2 of C4. In any case, C1 is uniquely embedable 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. By one example, we briefly discuss 2 − (n, 3, 2) multidesigns with similar restrictions. We also show a connection of the problem with the problem of completing latin hypercuboids of order 4 to latin hypercubes.