Extremal Type II $\mathbb{Z}_4$-codes are a class of self-dual $\mathbb{Z}_4$-codes with Euclidean weights divisible by eight and the largest possible minimum Euclidean weight for a given length. A small number of such codes is known for lengths greater than or equal to $48.$ The doubling method is a method for constructing Type II $\mathbb{Z}_4$-codes from a given Type II $\mathbb{Z}_4$-code. Based on the doubling method, in this paper we develop a method to construct new extremal Type II $\mathbb{Z}_4$-codes starting from an extremal Type II $\mathbb{Z}_4$-code of type $4^k$ with an extremal residue code and length $48, 56$ or $64$. Using this method, we construct three new extremal Type II $\mathbb{Z}_4$-codes of length $64$ and type $4^{31}2^2$. Extremal Type II $\mathbb{Z}_4$-codes of length $64$ of this type were not known before. Moreover, the residue codes of the constructed extremal $\mathbb{Z}_4$-codes are new best known $[64,31]$ binary codes and the supports of the minimum weight codewords of the residue code and the torsion code of one of these codes form self-orthogonal $1$-designs.

翻译：极值Type II $\mathbb{Z}_4$码是一类自对偶$\mathbb{Z}_4$码，其欧几里得权重能被8整除，且在给定长度下具有尽可能大的最小欧几里得权重。对于长度大于等于$48$的此类码，目前已知的数目很少。加倍方法是一种从给定Type II $\mathbb{Z}_4$码出发构造Type II $\mathbb{Z}_4$码的方法。基于加倍方法，本文发展了一种新方法，从具有极值剩余码且长度为$48$、$56$或$64$的$4^k$型极值Type II $\mathbb{Z}_4$码出发，构造新的极值Type II $\mathbb{Z}_4$码。利用该方法，我们构造了三个长度为$64$、类型为$4^{31}2^2$的新型极值Type II $\mathbb{Z}_4$码。此前，此类长度为$64$的极值Type II $\mathbb{Z}_4$码尚属未知。此外，所构造的极值$\mathbb{Z}_4$码的剩余码是新的最优$[64,31]$二元码，且其中一个码的剩余码和挠码的最小权重码字支撑集构成自正交$1$-设计。