Completely regular codes from regular spreads and linear SQS in Grassmann and Johnson graphs

It is known that every $(k-1)$-$(n,k,1)_q$ design $D$ and its induction is a completely regular code in the Grassmann graphs $J_q(n,k)$ and $J_q (n,k+1)$. It is shown that $D$ also induces a completely regular code in the graph $J_q(n,4)$ if $D$ is a regular spread in $J_q(n,2)$. Also, a new series of completely regular codes in the Johnson graph $J(n,6)$ is obtained from the Steiner quadruple system of the extended Hamming code. The talk will outline the basic elements of the proof and give the intersection arrays of completely regular codes of the Grassmann graph $G_2(6,3)$ of radius $1$.