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It is well known that space-time block codes (STBCs) obtained from orthogonal designs (ODs) are single-symbol decodable (SSD) and from quasi-orthogonal designs (QODs) are double-symbol decodable (DSD). However, there are SSD codes that are not obtainable from ODs and DSD codes that are not obtainable from QODs. In this paper, a method of constructing g-symbol decodable ( g-SD) STBCs using representations of Clifford algebras are presented which when specialized to g=1,2 gives SSD and DSD codes, respectively. For the number of transmit antennas 2a the rate (in complex symbols per channel use) of the g -SD codes presented in this paper is [(a+1-g)/(2a-g)]. The maximum rate of the DSD STBCs from QODs reported in the literature is [(a)/(2a-1)] which is smaller than the rate [(a-1)/(2a-2)] of the DSD codes of this paper, for 2a transmit antennas. In particular, the reported DSD codes for 8 and 16 transmit antennas offer rates 1 and 3/4 , respectively, whereas the known STBCs from QODs offer only 3/4 and 1/2, respectively. The construction of this paper is applicable for any number of transmit antennas. The diversity sum and diversity product of the new DSD codes are studied. It is shown that the diversity sum is larger than that of all known QODs and hence the new codes perform better than the comparable QODs at low signal-to-noise ratios (SNRs) for identical spectral efficiency. Simulation results for DSD codes at various spectral efficiencies are provided.