New families of completely transitive codes and distance transitive graphs

TL;DR

This paper introduces new infinite families of linear binary completely transitive codes with specific covering radii, which are used to construct infinite families of distance transitive graphs with diameters three and four.

Contribution

It presents novel infinite families of completely transitive codes derived from Hamming codes, and constructs corresponding distance transitive graphs.

Findings

Codes have covering radius 3 and 4.

Constructs infinite families of distance transitive graphs.

Links codes to graph symmetry properties.

Abstract

In this paper new infinite families of linear binary completely transitive codes are presented. They have covering radius $\rho = 3$ and 4, and are a half part of the binary Hamming and the binary extended Hamming code of length $n=2^m-1$ and $2^m$, respectively, where $m$ is even. From these new completely transitive codes, in the usual way, i.e., as coset graphs, new presentations of infinite families of distance transitive coset graphs of diameter three and four, respectively, are constructed.