| Summary: | We consider the problem of designing a low-complexity decoder for antipodal uniquely decodable (UD) /errorless code sets for overloaded synchronous code-division multiple access (CDMA) systems, where the number of signals <inline-formula> <tex-math notation="LaTeX">$K_{\mathrm{max}}^{a}$ </tex-math></inline-formula> is the largest known for the given code length <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>. In our complexity analysis, we illustrate that compared to maximum-likelihood (ML) decoder, which has an exponential computational complexity for even moderate code lengths, the proposed decoder has a quasi-quadratic computational complexity. Simulation results in terms of bit-error-rate (BER) demonstrate that the performance of the proposed decoder has only a <inline-formula> <tex-math notation="LaTeX">$1-2$ </tex-math></inline-formula> dB degradation in signal-to-noise ratio (SNR) at a BER of 10<sup>−3</sup> when compared to ML. Moreover, we derive the proof of the minimum Manhattan distance of such UD codes and we provide the proofs for the propositions; these proofs constitute the foundation of the formal proof for the maximum number users <inline-formula> <tex-math notation="LaTeX">$K_{\mathrm{max}}^{a}$ </tex-math></inline-formula> for <inline-formula> <tex-math notation="LaTeX">$L=8$ </tex-math></inline-formula>.
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