Completely regular codes in Johnson and Grassmann graphs with small covering radii

TL;DR

This paper constructs and analyzes new completely regular codes in Johnson and Grassmann graphs with small covering radii, expanding the understanding of their structure and properties.

Contribution

It introduces novel completely regular codes in Grassmann and Johnson graphs using spreads, Steiner systems, and linear programming methods.

Findings

Constructed a completely regular code in $J_q(n,4)$ from a Desarguesian 2-spread.

Developed a completely regular code in $J(n,6)$ from the Steiner quadruple system.

Identified several new completely regular codes with covering radius 1 in $J_2(6,3)$.

Abstract

Let L be a Desarguesian 2-spread in the Grassmann graph $J_q(n,2)$. We prove that the collection of the 4-subspaces, which do not contain subspaces from L is a completely regular code in $J_q(n,4)$. Similarly, we construct a completely regular code in the Johnson graph $J(n,6)$ from the Steiner quadruple system of the extended Hamming code. We obtain several new completely regular codes covering radius 1 in the Grassmann graph $J_2(6,3)$ using binary linear programming.