On a family of binary completely transitive codes with growing covering radius

TL;DR

This paper introduces a new family of binary linear completely transitive codes with increasing covering radius, leading to the construction of distance-transitive graphs with growing diameter, expanding the understanding of code and graph structures.

Contribution

The paper constructs a novel family of binary completely transitive codes with growing covering radius and demonstrates their connection to distance-transitive graphs.

Findings

Existence of codes with arbitrary covering radius r

Construction of codes with specific lengths for each r

Induction of distance-transitive graphs with increasing diameter

Abstract

A new family of binary linear completely transitive (and, therefore, completely regular) codes is constructed. The covering radius of these codes is growing with the length of the code. In particular, for any integer r > 1, there exist two codes with d=3, covering radius r and length 2r(4r-1) and (2r+1)(4r+1), respectively. These new completely transitive codes induce, as coset graphs, a family of distance-transitive graphs of growing diameter.