Code-Generating Factorizations of the $n$-Dimensional Unit Cube and of Perfect Binary Codes

In the paper, an approach to constructing and studying perfect binary codes with distance 3 is introduced. This approach is to study their modifications, i.e., perfect codes with the same projection along a given coordinate, and is based on studying corresponding connection structures over an $n$-cube and their factorizations, which are called code-generating. Basic properties of the notions mentioned are presented. Examples of their application to the construction of rather extensive classes of perfect codes are given.