Construction of extremal $ \mathbb{Z}_{4} $-codes using a neighborhood search algorithm

In this paper, we present a method for constructing extremal $ \mathbb{Z}_{4} $-codes based on random neighborhood search. This method is used to find new extremal Type Ⅰ and Type Ⅱ $ \mathbb{Z}_{4} $-codes of lengths 32 and 40. For the length 32, at least 182 new Type Ⅱ extremal $ \mathbb{Z}_{4} $-codes of types $ 4^{k}2^{32-2k} $, $ k\in\left\{9,10,12,13,14,15,16\right\} $ are constructed. In addition, we obtained at least 762 new extremal Type Ⅰ $ \mathbb{Z}_{4} $-codes of types $ 4^{k}2^{32-2k} $, $ k\in\left\{7,9,10,12,13,14,15,16\right\} $. For the length 40, constructed extremal $ \mathbb{Z}_{4} $-codes are of types $ 4^{k}2^{40-2k} $, $ k\in\left\{7,10,11,15,16\right\} $. There are at least 40 new Type Ⅱ extremal $ \mathbb{Z}_{4} $-codes, and at least 4144 new Type Ⅰ extremal $ \mathbb{Z}_{4} $-codes.