Three-weight codes over rings and strongly walk regular graphs

TL;DR

This paper constructs strongly walk-regular graphs from duals of three-weight codes over rings, including chain rings, using algebraic coding theory, and provides new insights into the nonlinear nature of Kerdock codes.

Contribution

It introduces a novel method to build strongly walk-regular graphs from three-weight codes over rings, expanding the connection between coding theory and graph theory.

Findings

Construction of strongly walk-regular graphs from codes over rings.

Infinite families derived from Kerdock and Teichmüller codes.

Alternative proof of the nonlinearity of Kerdock codes.

Abstract

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.