We give methods for constructing many self-dual $\mathbb{Z}_m$-codes and Type II $\mathbb{Z}_{2k}$-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 Type II $\mathbb{Z}_{2k}$-codes of length $32$ for $k=4,5,\ldots,10$. We also construct new extremal Type II $\mathbb{Z}_4$-codes of lengths $56$ and $64$.
翻译：我们用多种方法来构建许多自成一体的$\mathbb+m美元代码和二类的自成一体的$$\mathbb%2k}代码和二类的自成一体的$2n美元代码,分别从给定的自成一体的$\mathbb+m美元代码和二类的自成一体的$2n美元代码和二类的自成一体的$2n美元代码开始。作为一个应用,我们用美元=4,5,5,\ldots,20美元和二类的外形的2n美元代码来构建二类的自成二类的自成一体型代码。我们还用美元=4,5,\ldoldots,10美元来构建新的二类的外形二类 $\mathbbb+2k$24美元代码,56美元和64美元。
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