Construction of extremal Type II $\mathbb{Z}_{8}$-codes via doubling method

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.