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.