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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 largestpossible minimum Euclidean weight for a given length. A small number of suchcodes is known for lengths greater than or equal to $48.$ The doubling methodis 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 amethod to construct new extremal Type II $\mathbb{Z}_4$-codes starting from anextremal Type II $\mathbb{Z}_4$-code of type $4^k$ with an extremal residuecode and length $48, 56$ or $64$. Using this method, we construct three newextremal 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 notknown before. Moreover, the residue codes of the constructed extremal$\mathbb{Z}_4$-codes are new best known $[64,31]$ binary codes and the supportsof the minimum weight codewords of the residue code and the torsion code of oneof these codes form self-orthogonal $1$-designs.

New extremal Type II $\mathbb{Z}_4$-codes of length 64 by the doubling method