In this letter, an uplink communication scenario in which one dual-antenna transmitter communicates with a receiver having massive antennas is considered. We propose a $2\times 2$ space-time transmission scheme, which does not require any knowledge of instantaneous channel state information at either transmitter or receiver. Based on the non-coherent maximum-likelihood detector, we derive the pairwise error probability (PEP) performance and analyze its upper and lower bounds, which show the PEP decays exponentially with the number of receiving antennas. In addition, two criteria are obtained for designing the transmitting signal matrix ${\mathbf{S}}$ , i.e., the determinant and trace criteria of matrix ${\mathbf{S}}^{H}{\mathbf{S}}$ . With these criteria, we present a space-time signal design based on PSK constellation. Finally, computer simulations are performed to show how this design works, and numerical results validate the tightness of the upper and lower bounds when the number of receiving antennas is large.