In this paper, we consider sparse index modulation (SIM) for precoded orthogonal frequency division multiplexing (OFDM) and its generalization to multi-carrier multiple access systems. The resulting multiple access is called sparse index multiple access (SIMA). SIMA can be seen as a combination of multi-carrier code division multiple access (MC-CDMA) and SIM. Thus, SIMA is able to exploit a path diversity gain by (random) spreading over multiple carriers as MC-CDMA. To detect multiple users’ signals, a low-complexity detection method is proposed by exploiting the notion of compressive sensing (CS). The derived low-complexity detection method is based on the orthogonal matching pursuit (OMP) algorithm, which is one of greedy algorithms used to estimate sparse signals in CS. From simulation results, we can observe that SIMA can perform better than MC-CDMA when the ratio of the number of users to the number of multi-carrier is low.