On the Existence of Some Completely Regular Codes in Hamming Graphs

We solve some first questions in the table of small parameters of completely regular (CR) codes in Hamming graphs
$H(n,\ q)$
. The most uplifting result is the existence of a {13, 6, 1; 1, 6, 9}-CR code in
$H(n,\ 2), n \geq$
13. We also establish the non-existence of a {II, 4; 3, 6}-code and a {10, 3; 4, 7}-code in
$H(12,2)$
and
$H(13,2)$
. A partition of the complement of the quaternary Hamming code of length 5 into 4-cliques is found, which can be used to construct completely regular codes with covering radius 1 by known constructions. Additionally we discuss the parameters {24, 21, 10; 1,4, 12} of a putative completely regular code in
$H(24,2)$
and show the nonexistence of such a code in
$H(8,4)$
.