  Interference alignment is known to achieve the maximum sum DoF of
$\frac{4M}{3}$ in the $2 \times 2$ $X$-Network (i.e., two-transmitter (Tx)
two-receiver (Rx) $X$-Network) with $M$ antennas at each node, as demonstrated
by Jafar and Shamai. Recently, an Alamouti code based transmission scheme,
which we call the Li-Jafarkhani-Jafar (LJJ) scheme, was proposed for the $2
\times 2$ $X$-Network with two antennas at each node. This scheme achieves 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. In the LJJ scheme,
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
$\frac{8}{3}$. Bit error rate (BER) is an important performance metric when the
coding length is finite. This work first proposes a new STBC for a three
transmit antenna single user MIMO system. Building on this STBC, we extend the
LJJ scheme to the $2 \times 2$ $X$-Network with three antennas at each node.
Local channel knowledge is assumed at each Tx. It is shown that the proposed
scheme achieves the maximum possible sum DoF of 4. A diversity gain of 3 is
also guaranteed when fixed finite constellation inputs are used.
