Joint communication and sensing (JCAS) is a very promising 6G technology, which attracts more and more research attention. Compared with communication, radar has many unique features in terms of waveform design criteria, self-interference cancellation (SIC), aperture-dependent resolution, and virtual aperture. This paper proposes a novel waveform design named max-aperture radar slicing (MaRS) to gain a large time-frequency aperture, which is generated by orthogonal frequency division multiplexing (OFDM) and occupies only a tiny fraction of OFDM resources. The proposed MaRS keeps the radar advantages of constant modulus, zero auto-correlation sequence, and simple SIC. As MaRS consumes much less resources, conventional processing methods fail, and novel angle-Doppler map based methods are proposed to obtain the range-velocity-angle information from MaRS echos and strong clutters. To avoid complex full-duplex communication, this paper proposes a hybrid-duplex JCAS scheme composed of half-duplex communication and full-duplex radar. The half-duplex communication antenna array is reused, and a small sensing-dedicated antenna array is added. Using these two arrays, a large space-domain sensing aperture is virtually formed to greatly improve the angle resolution. The numerical results show that the proposed MaRS and hybrid duplex can achieve a high sensing resolution with only 0.4% OFDM resources, which reduces the overheads of conventional methods to less than one tenth.