The upcoming 5G networks demand high-speed and high spectral-efficiency communications to keep up with the proliferating traffic demands. To this end, Massive multiple-input multiple-output (MIMO) techniques have gained significant traction owing to its ability to achieve these without increasing bandwidth or density of base stations. The preexisting space-time block code (STBC) designs cannot achieve a rate of more than 1 for more than two transmit antennas while preserving the orthogonality and full diversity conditions. In this paper, we present Jagannath codes - a novel complex modulation STBC, that achieves a very high rate of 2 for three and four transmit antennas. The presented designs achieve full diversity and overcome the previously achieved rates with the three and four antenna MIMO systems. We present a detailed account of the code construction of the proposed designs, orthogonality and full diversity analysis, transceiver model and conditional maximum likelihood (ML) decoding. In an effort to showcase the improvement achieved with the presented designs, we compare the rates and delays of some of the known STBCs with the proposed designs. The effective spectral efficiency and coding gain of the presented designs are compared to the Asymmetric Coordinate Interleaved design (ACIOD) and Jafarkhani code. We presented an effective spectral efficiency improvement by a factor of 2 with the proposed Jagannath codes. Owing to the full diversity of the presented designs, we demonstrate significant coding gains (6 dB and 12 dB) with the proposed designs.