Completely Regular Codes in Johnson and Grassmann Graphs with Small Covering Radii

Abstract

Let ${\cal L}$ 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 ${\cal L}$ 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 with covering radius $1$ in the Grassmann graph $J_2(6,3)$ using binary linear programming.