On the residue codes of extremal Type II ℤ ₄-codes of lengths 32 and 40

In this paper, we determine the dimensions of the residue codes of extremal Type II ^Z4-codes of lengths 32 and 40. We demonstrate that every binary doubly even self-dual code of length 32 can be realized as the residue code of some extremal Type II ^Z4-code. It is also shown that there is a unique extremal Type II ^Z4-code of length 32 whose residue code has the smallest dimension 6 up to equivalence. As a consequence, many new extremal Type II ^Z4-codes of lengths 32 and 40 are constructed.