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We give methods for constructing many self-dual $\mathbb{Z}_m$-codes and TypeII $\mathbb{Z}_{2k}$-codes of length $2n$ starting from a given self-dual$\mathbb{Z}_m$-code and Type II $\mathbb{Z}_{2k}$-code of length $2n$,respectively. As an application, we construct extremal Type II$\mathbb{Z}_{2k}$-codes of length $24$ for $k=4,5,\ldots,20$ and extremal TypeII $\mathbb{Z}_{2k}$-codes of length $32$ for $k=4,5,\ldots,10$. We alsoconstruct new extremal Type II $\mathbb{Z}_4$-codes of lengths $56$ and $64$.

Construction of extremal Type II $\mathbb{Z}_{2k}$-codes