In this letter, a non-orthogonal multiple access system based on combinatorial designs (C-NOMA) is studied. The sparse pattern of balanced incomplete block designs provide an inherent mapping for this system. A novel constellation design is proposed to avoid the indeterminate equations when multiuser data is linearly combined over the resources. It can form a bijection mapping with much lower complexity compared to other NOMA approaches that require the design of complex codebooks and decoders. This will allow a larger number of users. The simulation results show that our system obtains an equivalent and/or better performance compared to other NOMA systems.