Joint communication and sensing (JCAS) is a very promising 6G technology, which attracts more and more research attention. Unlike communication, radar has many unique features in terms of waveform criteria, self-interference cancellation (SIC), aperture-dependent resolution, and virtual aperture. This paper proposes a waveform design named max-aperture radar slicing (MaRS) to gain a large time-frequency aperture, which reuses the orthogonal frequency division multiplexing (OFDM) hardware and occupies only a tiny fraction of OFDM resources. The proposed MaRS keeps the radar advantages of constant modulus, zero auto-correlation, and simple SIC. Joint space-time processing algorithms are proposed to recover the range-velocity-angle information from strong clutters. Furthermore, this paper proposes a hybrid-duplex JCAS scheme where communication is half-duplex while radar is full-duplex. In this scheme, the half-duplex communication antenna array is reused, and a small sensing-dedicated antenna array is specially designed. Using these two arrays, a large space-domain aperture is virtually formed to greatly improve the angle resolution. The numerical results show that the proposed MaRS and hybrid-duplex schemes achieve a high sensing resolution with less than 0.4% OFDM resources and gain an almost 100% hit rate for both car and UAV detection at a range up to 1 km.