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

Abstract

We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary $(n=2^m-3,2^{n-m-1},4)$ code $C$, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of an equitable partition of the $n$-cube into six cells. An arbitrary binary $(n=2^m-4,2^{n-m},3)$ code $D$, i.e., a code with parameters of a triply-shortened Hamming code, is a cell of an equitable family (but not a partition) from six cells. As a corollary, the codes $C$ and $D$ are completely semiregular; i.e., the weight distribution of such a code depends only on the minimal and maximal codeword weights and the code parameters. Moreover, if $D$ is self-complementary, then it is completely regular. As an intermediate result, we prove, in terms of distance distributions, a general criterion for a partition of the vertices of a graph (from rather general class of graphs, including the distance-regular graphs) to be equitable. Keywords: 1-perfect code; triply-shortened 1-perfect code; equitable partition; perfect coloring; weight distribution; distance distribution

12 pages

References in corpus (4)

• On weight distributions of perfect colorings and completely regular codes
• On the binary codes with parameters of doubly-shortened 1-perfect codes
• On Optimal Binary One-Error-Correcting Codes of Lengths $2^m-4$ and $2^m-3$
• Two Optimal One-Error-Correcting Codes of Length 13 That Are Not Doubly Shortened Perfect Codes

Cited by in corpus (8)

• 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 multifold packings of radius-1 balls in Hamming graphs
• On calculation of the interweight distribution of an equitable partition
• On $q$-ary shortened-$1$-perfect-like codes
• Perfect colorings of the infinite square grid: coverings and twin colors