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We show that regular homogeneous two-weight $\mathbb{Z}_{p^k}$-codes where$p$ is odd and $k\geqslant 2$ with dual Hamming distance at least four do notexist. The proof relies on existence conditions for the strongly regular graphbuilt on the cosets of the dual code.

Two weight $\mathbb{Z}_{p^k}$-codes, $p$ odd prime