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In this treatise, we present the concept of compressed-sensing (CS)-aided space-time shift keying index modulation (STSK-IM), where a virtual domain orthogonal frequency division multiplexing (OFDM) symbol is divided into Na-sized blocks, which carry K STSK codewords over a specific combination of virtual-domain sub-carriers and then converted to the frequency domain with the aid of a CS matrix. We design the system for operation in the milli-meter wave (mmWave) frequency band. Furthermore, we amalgamate our soft-decision-aided scheme both with the concept of coordinate-interleaving as well as a discrete Fourier transform-aided codebook design conceived for analogue beamforming, which is vitally important for mmWave systems. In the proposed system, the number of implicit bits conveyed by the activated sub-carrier frequency index (FI) is determined by the number of the available K to N_a subcarrier permutations. Hence, we propose two FI allocation techniques, namely the distinct FI and the shared FI-based schemes, which strike a tradeoff between the attainable sparsity level and the achievable capacity limit. We then introduce a reduced-complexity detection technique in order to mitigate the detection complexity order of the optimum detector from O(N_c · (Q · L)^K) to O(N̂_c · (Q · L)^K), whereN̂_c ≤ N_c. We also formulate the discrete-input continuous-output memoryless channel capacity and invoke EXtrinsic Information Transfer charts for characterizing the achievable performance limit of the reduced-complexity aided detector. Finally, we analyze the bit error rate performance of both the uncoded and of our coded CS-aided STSK-IM systems associated with both the optimum and the reduced-complexity detectors.