Space-Time Block Codes (STBCs) suffer from a prohibitively high decoding complexity unless the low-complexity decodability property is taken into consideration in the STBC design. For this purpose, several families of STBCs that involve a reduced decoding complexity have been proposed, notably the multi-group decodable and the fast decodable (FD) codes. Recently, a new family of codes that combines both of these families namely the fast group decodable (FGD) codes was proposed. In this paper, we propose a new construction scheme for rate-1 FGD codes for 2^a transmit antennas. The proposed scheme is then applied to the case of four transmit antennas and we show that the new rate-1 FGD code has the lowest worst-case decoding complexity among existing comparable STBCs. The coding gain of the new rate-1 code is optimized through constellation stretching and proved to be constant irrespective of the underlying QAM constellation prior to normalization. Next, we propose a new rate-2 FD STBC by multiplexing two of our rate-1 codes by the means of a unitary matrix. Also a compromise between rate and complexity is obtained through puncturing our rate-2 FD code giving rise to a new rate-3/2 FD code. The proposed codes are compared to existing codes in the literature and simulation results show that our rate-3/2 code has a lower average decoding complexity while our rate-2 code maintains its lower average decoding complexity in the low SNR region. If a time-out sphere decoder is employed, our proposed codes outperform existing codes at high SNR region thanks to their lower worst-case decoding complexity.