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

TL;DR

This paper investigates the structure of certain binary codes related to 1-perfect codes, showing how they can be embedded into larger codes and characterizing their partitioning properties within the n-cube.

Contribution

It establishes that binary doubly-shortened 1-perfect codes are part of an equitable partition of the n-cube and characterizes conditions for their extension to larger 1-perfect codes.

Findings

Any such code is part of an equitable partition of the n-cube.

Extension to a larger 1-perfect code depends on splitting a specific part into two distance-3 codes.

Codes are uniquely embeddable in a twofold 1-perfect code with structural restrictions.

Abstract

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.