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Space-Time Block Coded (STBC) Orthogonal Frequency Division Multiplexing (OFDM) satisfies higher data-rate requirements while maintaining signal quality in a multipath fading channel. However, conventional STBCs, including Orthogonal STBCs (OSTBCs), Non-Orthogonal (NOSTBCs), and Quasi-Orthogonal STBCs (QOSTBCs), do not provide both maximal diversity order and unity code rate simultaneously for more than two transmit antennas. This paper targets this problem and applies Maximum Rank Distance (MRD) codes in designing STBC-OFDM systems. By following the direct-matrix construction method, we can construct binary extended finite field MRD-STBCs for any number of transmitting antennas. Work uses MRD-STBCs built over Phase-Shift Keying (PSK) modulation to develop an MRD-based STBC-OFDM system. The MRD-based STBC-OFDM system sacrifices minor error performance compared to traditional OSTBC-OFDM but shows improved results against NOSTBC and QOSTBC-OFDM. It also provides 25% higher data-rates than OSTBC-OFDM in configurations that use more than two transmit antennas. The tradeoffs are minor increases in computational complexity and processing delays.