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

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.

12pp

References in corpus (2)

• n-Ary quasigroups of order 4
• Two Optimal One-Error-Correcting Codes of Length 13 That Are Not Doubly Shortened Perfect Codes

Cited by in corpus (8)

• On weight distributions of perfect colorings and completely regular codes
• Perfect 2-colorings of Hamming graphs
• On the OA(1536,13,2,7) and related orthogonal arrays
• The extended 1-perfect trades in small hypercubes
• On Optimal Binary One-Error-Correcting Codes of Lengths $2^m-4$ and $2^m-3$
• On the binary codes with parameters of triply-shortened 1-perfect codes
• On multifold packings of radius-1 balls in Hamming graphs
• On $q$-ary shortened-$1$-perfect-like codes