Non-orthogonal multiple-access (NOMA) is designed to transmit massive amounts of user communications. The incidence matrix manages the relationship between users and resources. This study focused on increasing user supportability and complexity reduction using larger incidence matrices. Our approach to optimizing the incidence matrix to improve system capacity and reduce complexity is based on two critical mathematical concepts: Parallel classes in hypergraph theory and combinatorial designs allow us to explore and extend incidence matrices. Frame theory is used to estimate matrix structures. Then, we investigate applications utilizing our incidence matrix designs-Simple Orthogonal Multi-Arrays (SOMA). The characteristics of SOMA reflect the unique Latin Square pattern, allowing us to produce a highly flexible and fair resource allocation matrix. SOMA designs let us support overload factors over 500%. The theoretical performance analysis equations of our NOMA system were established to support dynamic adaptability and optimization. We implemented and evaluated security methods for eavesdroppers. The prototype of the user hierarchy allows a higher-priority group to have a lower error rate without significantly affecting the system's performance. Finally, the Monte Carlo simulation indicated that our NOMA systems allow higher degrees of freedom and lower complexity than other NOMA schemes while maintaining graceful error rates with a maximum 33% improvement.INDEX TERMS Incidence matrix, NOMA, combinatorial design, frame theory, SOMA.

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