Light fidelity (LiFi), which occupies part of the optical spectrum, is a promising technology to meet the high-speed wireless data demand. For an additional throughput gain along with low energy consumption, index modulation (IM) has begun to be combined with LiFi systems. The main advantage of IM is its ability to encode information in both a conventional transmit signal as well as the index of the transmit entity such as through the transmitter, subcarrier and time-slot domain indexes. Time-slot IM (TIM) offers additional degrees of freedom to single-carrier (SC) single-input-single-output systems to encode information onto pulse activation patterns. In this paper, we propose a generalized TIM (GTIM) transmission method for SC frequency domain equalization (SC-FDE)-based systems to fully harness the benefits of TIM. In GTIM, both the number and locations of active time-slots are flexible such that they are different for each transmit symbol. Moreover, the active time-slots carry pulse amplitude modulation (PAM) symbols in the signal domain. The proposed system also benefits from SC-FDE which simplifies the transmission and detection complexity at both sides. We also propose a set partitioning algorithm that chooses the optimal transmission symbols set and labels them concurrently. The bit error ratio (BER) performance of GTIM is compared to conventional PAM and TIM under a limited dynamic range constraint and frequency-selectivity. It is shown that the proposed GTIM achieves up to a 3 dB BER performance gain.