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

Type II $ \mathbb{Z}_{8} $-codes are a class of self-dual codes over $ \mathbb{Z}_{8} $ with Euclidean weights divisible by $ 16. $ A Type II $ \mathbb{Z}_{8} $-code of length $ n $ is extremal if its minimum Euclidean weight is equal to $ 16\left\lfloor\frac{n}{24} \right\rfloor+ 16. $ We introduce the doubling method for constructing a Type II $ \mathbb{Z}_{2k} $-code of length $ n $ from a known Type II code of the same length. Based on this method, we develop an algorithm to construct new extremal Type II $ \mathbb{Z}_8 $-codes starting from a known 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.