Families of nested completely regular codes and distance-regular graphs

TL;DR

This paper constructs infinite families of nested completely regular binary codes with specific covering radii, which lead to the creation of distance-regular and distance-transitive coset graphs, advancing the understanding of code and graph structures.

Contribution

It introduces new infinite families of nested completely regular codes derived from Hamming codes, and constructs associated distance-regular and distance-transitive graphs.

Findings

Constructed codes have covering radius 3 or 4.

Generated embedded distance-regular coset graphs.

Some codes are completely transitive, with graphs being distance-transitive.

Abstract

In this paper infinite families of linear binary nested completely regular codes are constructed. They have covering radius $\rho$ equal to $3$ or $4$, and are $1/2^i$-th parts, for $i\in\{1,\ldots,u\}$ of binary (respectively, extended binary) Hamming codes of length $n=2^m-1$ (respectively, $2^m$), where $m=2u$. In the usual way, i.e., as coset graphs, infinite families of embedded distance-regular coset graphs of diameter $D$ equal to $3$ or $4$ are constructed. In some cases, the constructed codes are also completely transitive codes and the corresponding coset graphs are distance-transitive.