Three-weight codes over rings and strongly walk regular graphs

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.