  A transmission scheme based on the Alamouti code, which we call the
Li-Jafarkhani-Jafar (LJJ) scheme, was recently proposed for the $2 \times 2$
$X$ Network (i.e., two-transmitter (Tx) two-receiver (Rx) $X$ Network) with two
antennas at each node. This scheme was claimed to achieve a sum degrees of
freedom (DoF) of $\frac{8}{3}$ and also a diversity gain of two when fixed
finite constellations are employed at each Tx. Furthermore, each Tx required
the knowledge of only its own channel unlike the Jafar-Shamai scheme which
required global CSIT to achieve the maximum possible sum DoF of $\frac{8}{3}$.
In this paper, we extend the LJJ scheme to the $2 \times 2$ $X$ Network with
four antennas at each node. The proposed scheme also assumes only local channel
knowledge at each Tx. We prove that the proposed scheme achieves the maximum
possible sum DoF of $\frac{16}{3}$. In addition, we also prove that, using any
fixed finite constellation with appropriate rotation at each Tx, the proposed
scheme achieves a diversity gain of at least four.
