  Let $R=\mathbb{Z}_4$ be the integer ring mod $4$. A double cyclic code of
length $(r,s)$ over $R$ is a set that can be partitioned into two parts that
any cyclic shift of the coordinates of both parts leaves invariant the code.
These codes can be viewed as $R[x]$-submodules of $R[x]/(x^r-1)\times
R[x]/(x^s-1)$. In this paper, we determine the generator polynomials of this
family of codes as $R[x]$-submodules of $R[x]/(x^r-1)\times R[x]/(x^s-1)$.
Further, we also give the minimal generating sets of this family of codes as
$R$-submodules of $R[x]/(x^r-1)\times R[x]/(x^s-1)$. Some optimal or suboptimal
nonlinear binary codes are obtained from this family of codes. Finally, we
determine the relationship of generators between the double cyclic code and its
dual.
