Multicarrier waveforms have always been of high importance in many radar and communication applications as they have several desirable properties. Multicarrier Direct Sequence-CDMA (MC-DS-CDMA) is a well known modulation approach. It can be adapted to joint radar-communication systems with appropriate design choices. For the radar sensing purposes it is desirable to have ambiguity function resembling the ideal thumbtack function. In this paper, we propose a joint radar-communication waveform stemming from MC-DS-CDMA. A subset of P subcarriers are dedicated to communications using orthogonal variable spreading factor codes and a scrambling code to facilitate different rates and reduced correlation among subcarrier signals. Radar subsystem uses spreading codes for ranging with desirable auto- and cross-correlation properties and a scrambling code to separate the radar and communication signals. It is demonstrated that the proposed waveforms have highly desirable ambiguity functions (AF). We study the peak-to-average power ratios of the waveforms, too.