On the SIC-ML Equivalence Under Superincreasing Power Allocation in NOMA

ForK -user binary phase-shift-keying (BPSK) non-orthogonal multiple access (NOMA) with successive interference cancellation (SIC), the power allocation (PA)α =(α _1,⋯,αK) determines both decodability and decoder complexity. We prove that whenα is superincreasing -√ α _k> ∑ _j> k√ α j for allk=1,⋯,K-1 - the per-symbolO(K) SIC decoder is realization-by-realization identical to the maximum-likelihood (ML) decoder over the 2K composite-constellation hypotheses. The proof exploits the Merkle-Hellman knapsack property: bit-by-bit greedy decoding from the strongest user uniquely identifies the closest constellation point. The result eliminates the standard SIC-vs-ML complexity gap when the designer can choose the PA, motivates a margin-parameterized superincreasing PA family with single design parameterϵ > 0 , and establishes that the optimal margin minimizing Bob's bit-error rate (BER) isϵ∗ ≈ 1 atK=2 and ϵ ∗ ≈ 0.5 atK=4. Monte Carlo simulations confirmϵ ∗ =0.5 delivers a 20 dB BER advantage over geometric PA atK=4 , SNR=35 dB, and that SIC and ML produce identical PbBob to four decimal places at every operating point, validating the theory.