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

TL;DR

This paper investigates the properties of binary codes with parameters similar to triply-shortened 1-perfect codes, revealing their equitable partition structures, semiregularity, and conditions for complete regularity.

Contribution

It characterizes the equitable partition and regularity properties of codes with parameters of triply-shortened 1-perfect codes, including new criteria for equitable partitions in graphs.

Findings

Codes are completely semiregular, with weight distributions depending only on minimal and maximal weights.

Self-complementary codes among these are completely regular.

Provides a general criterion for equitable partitions in graphs, including distance-regular graphs.

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