Three-Weight Codes over Rings and Strongly Walk Regular Graphs

We construct strongly walk-regular graphs as coset of the duals codes with three non-zero homogeneous weights over $${\mathbb {Z}}_{p^m},$$ for p a prime, and more generally chain rings depth m, residue field size q, prime power. In case $$p=m=2,$$ strong necessary conditions ( diophantine equations) on weight distribution are derived, leading to partial classification in modest length. Infinite families examples built from Kerdock generalized Teichmüller codes. As byproduct, we give an alternative proof that code is nonlinear.