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A new five-dimensional (5D) hyperchaotic system is presented in this paper. Four equations of the system each contain a cubic product term. We prove that the system is a true hyperchaotic system and has complex nonlinear dynamic behavior by computing and analyzing the chaotic attractor, Lyapunov exponents (LEs), bifurcation diagram, Poincare section and time domain waveform. Then, chaotic sequences are generated based on the new system and applied in a direct sequence code division multiple access (DS-CDMA) system. The randomness, correlation and anti-MAI (multiple access interference) are tested and analyzed through numeric computing and simulation. The results prove that the sequences can improve the performance of the DS-CDMA system effectively because they have good randomness, correlation and anti-MAI properties.