Extremal Type II $\mathbb{Z}_{8}$-codes are a class of self-dual $\mathbb{Z}_{8}$-codes with Euclidean weights divisible by $16$ and the largest possible minimum Euclidean weight for a given length. We introduce a doubling method for constructing a Type II $\mathbb{Z}_{2k}$-code of length $n$ from a known Type II $\mathbb{Z}_{2k}$-code of length $n$. Based on this method, we develop an algorithm to construct new extremal Type II $\mathbb{Z}_8$-codes starting from an extremal Type II $\mathbb{Z}_8$-code of type $(\frac{n}{2},0,0)$ with an extremal $\mathbb{Z}_4$-residue code and length $24, 32$ or $40$. We construct at least ten new extremal Type II $\mathbb{Z}_8$-codes of length $32$ and type $(15,1,1)$. Extremal Type II $\mathbb{Z}_8$-codes of length $32$ of this type were not known before. Moreover, the binary residue codes of the constructed extremal $\mathbb{Z}_8$-codes are optimal $[32,15]$ binary codes.

翻译：极大型II $\mathbb{Z}_{8}$-码是一类自对偶$\mathbb{Z}_{8}$-码，其欧几里得权重可被$16$整除，且在给定长度下具有最大可能的最小欧几里得权重。我们引入一种加倍方法，用于从已知长度为$n$的II型$\mathbb{Z}_{2k}$-码构造长度为$n$的II型$\mathbb{Z}_{2k}$-码。基于该方法，我们开发了一种算法，从具有极大型$\mathbb{Z}_4$-剩余码且长度为$24$、$32$或$40$的类型$(\frac{n}{2},0,0)$的极大型II $\mathbb{Z}_8$-码出发，构造新的极大型II $\mathbb{Z}_8$-码。我们构造了至少十个新的长度为$32$、类型为$(15,1,1)$的极大型II $\mathbb{Z}_8$-码。此前，该类型长度为$32$的极大型II $\mathbb{Z}_8$-码尚未被已知。此外，所构造的极大型$\mathbb{Z}_8$-码的二进制剩余码是最优的$[32,15]$二进制码。