A submodule, \(\Phi\) , of \(\mathbb {Z}_4^n\) is called a double cyclic code of length \(n=k+l\) over \(\mathbb {Z}_4\) if any cyclic shift of the first k coordinates and last l coordinates of a codeword is a codeword. The code \(\Phi\) is called a separable code, if \(\Phi =\Phi _k \times \Phi _l\) where \(\Phi _k\) is the canonical projection of \(\Phi\) on the first k coordinates and \(\Phi _l\) on the last l coordinates. If there exists a basis for \(\Phi\) , then \(\Phi\) is called free. Also, the code \(\Phi\) is a self-dual code, if \(\Phi =\Phi ^{\perp }\) . The self-dual code \(\Phi\) is Type II if the Euclidean weight of every codeword is divisible by 8, otherwise is Type I. In this paper, we study free double cyclic self-dual codes of length \(n=k+l\) over \(\mathbb {Z}_4\) . We investigate the separable free double cyclic codes of length \(n=k+l\) over \(\mathbb {Z}_4\) and show that the separable free double cyclic codes are not self-dual codes. Moreover, we provide the process of finding the nonseparable free double cyclic self-dual codes of length \(n=k+l\) over \(\mathbb {Z}_4\) . In particular, we determine \(n=8\) is the shortest length of the codes. Finally, we classify Type I and Type II codes of the nonseparable self-dual codes and study optimal codes of them.
