  In this paper infinite families of linear binary nested completely regular
codes are constructed. They have covering radius $\rho$ equal to $3$ or $4$,
and are $1/2^i$-th parts, for $i\in\{1,\ldots,u\}$ of binary (respectively,
extended binary) Hamming codes of length $n=2^m-1$ (respectively, $2^m$), where
$m=2u$. In the usual way, i.e., as coset graphs, infinite families of embedded
distance-regular coset graphs of diameter $D$ equal to $3$ or $4$ are
constructed. In some cases, the constructed codes are also completely
transitive codes and the corresponding coset graphs are distance-transitive.
