  We construct strongly walk-regular graphs as coset graphs of the duals of
codes with three non-zero homogeneous weights over $\mathbb{Z}_{p^m},$ for $p$
a prime, and more generally over chain rings of depth $m$, and with a residue
field of size $q$, a prime power. Infinite families of examples are built from
Kerdock and generalized Teichm\"uller codes. As a byproduct, we give an
alternative proof that the Kerdock code is nonlinear.
