In this paper, we investigate the structural properties of hulls associated with non-separable double cyclic codes (NSDCCs) over the quaternary ring \(\mathbb {Z }_4 \) . Firstly, we analyze the generators of these hulls and derive a necessary condition for NSDCCs over \(\mathbb {Z }_4\) to be linear complementary dual (LCD). This condition enables the construction of several LCD codes over \(\mathbb {Z }_4\) with previously undocumented parameters. Subsequently, we establish formulas to determine the 2-dimensions of hulls for NSDCCs and obtain the enumeration of such codes whose hulls having a fixed 2-dimension. Finally, we compute the average 2-dimension of hulls of NSDCCs over \(\mathbb {Z}_4\) .
