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A suboptimum search algorithm to suppress the interference level in conjunction with the primary stage of the polynomial expanded (PE) linear multiuser detector for the mobile uplink is proposed. The initial stage is improved through mathematical analysis via Rayleigh-Ritz and Gershgorin algorithms in the linear algebra and based on them PER and PEG detectors are introduced. Its performance is also compared with other recently reported methods in the literature, amongst which is a comprehensive method that optimises the weights of the Taylor series asymptotically. The complexity of the initial stage is also evaluated and compared to the recently reported low-complex Fourier interference cancellation method. Depending on the value of the spreading factor of the active users in system, PEG outperforms the Fourier algorithm (in terms of complexity) with a large gap. The technique was compared with suboptimum techniques that result in best performance and also compared in a DS-CDMA with less complex techniques such as the Fourier interference cancellation method. The Fourier algorithm is also investigated and some comments on its applicability in the FDD DS-CDMA are also made. The structure of the PE method and the suboptimum search algorithm are well suited together and that makes them collaboratively work without encountering a high level of complexity. Considering the power profile of the users in the suboptimum search algorithm leads to even less complexity, yet keeps the performance almost the same. The performance of the structure is obtained by simulations and has been compared to the partial parallel interference cancellation (PPIC) method. A good improvement in performance in the low SNR regions, which is difficult to achieve by conventional multiuser detectors and is also important as the actual systems are likely to operate in these regions, has been achieved. All the techniques and their modifications introduced in this work consider the complexity as an important issue that make them suitable for industry and implementation purposes. Another important feature is that the techniques perform on canonical matrix formulations of the system so they can be applied to MC-CDMA and MIMO systems, as well.