Modern mobile terminals often produce a large number of small data packets. For these packets, it is inefficient to follow the conventional medium access control protocols because of poor utilization of service resources. We propose a novel multiple access scheme that employs block-spreading based precoding at the transmitters and sparsity-aware detection schemes at the base station. The proposed scheme is well suited for the emerging massive multiple-input multiple-output (MIMO) systems, as well as conventional cellular systems with a small number of base-station antennas. The transmitters employ precoding in time domain to enable the simultaneous transmissions of many users, which could be even more than the number of receive antennas at the base station. The system is modeled as a linear system of equations with block-sparse unknowns. We first adopt the block orthogonal matching pursuit (BOMP) algorithm to recover the transmitted signals. We then develop an improved algorithm, named interference cancellation BOMP (ICBOMP), which takes advantage of error correction and detection coding to perform perfect interference cancellation during each iteration of BOMP algorithm. Conditions for guaranteed data recovery are identified. The simulation results demonstrate that the proposed scheme can accommodate more simultaneous transmissions than conventional schemes in typical small-packet transmission scenarios.