Dual-function radar communication (DFRC) systems implement both sensing and communication using the same hardware. Such schemes are often more efficient in terms of size, power, and cost, over using distinct radar and communication systems. Since these functionalities share resources such as spectrum, power, and antennas, DFRC methods typically entail some degradation in both radar and communication performance. In this work we propose a DFRC scheme based on the carrier agile phased array radar (CAESAR), which combines frequency and spatial agility. The proposed DFRC system, referred to as multi-carrier agile joint radar communication (MAJoRCom), exploits the inherent spatial and spectral randomness of CAESAR to convey digital messages in the form of index modulation. The resulting communication scheme naturally coexists with the radar functionality, and thus does not come at the cost of reduced radar performance. We analyze the performance of MAJoRCom, quantifying its achievable bit rate. In addition, we develop a low complexity decoder and a codebook design approach, which simplify the recovery of the communicated bits. Our numerical results demonstrate that MAJoRCom is capable of achieving a bit rate which is comparable to utilizing independent communication modules without affecting the radar performance, and that our proposed low-complexity decoder allows the receiver to reliably recover the transmitted symbols with an affordable computational burden.