The performance of an asynchronous code division multiple access (CDMA) system (uplink) employing very low-rate maximum free distance codes for combined coding and spreading is analyzed when successive or parallel interference cancellation is applied. An analytical approach to the evaluation of the bit error rate is presented and shown to give results close to simulations. Our results show that the code-spread system outperforms the conventionally coded and spread system. Without interference cancellation the single-user bound is never reached (except for one user). With two stages of parallel interference cancellation, a code-spread system with a load only slightly less than 1 bit/chip can obtain a bit error rate very close to that of a single-user system.