Three-Weight Codes over Rings and Strongly Walk Regular Graphs

Titelangaben

Shi, Minjia ; Kiermaier, Michael ; Kurz, Sascha ; Solé, Patrick:
Three-Weight Codes over Rings and Strongly Walk Regular Graphs.
In: Graphs and Combinatorics. Bd. 38 (2022) . - 56.
ISSN 1435-5914
DOI: https://doi.org/10.1007/s00373-021-02430-6

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Abstract

We construct strongly walk-regular graphs as coset graphs of the duals of codes with three non-zero homogeneous weights over 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. In the case of p=m=2, strong necessary conditions (diophantine equations) on the weight distribution are derived, leading to a partial classification in modest length. Infinite families of examples are built from Kerdock and generalized Teichmüller codes. As a byproduct, we give an alternative proof that the Kerdock code is nonlinear.