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Abstract Periodic/aperiodic sequences with low autocorrelation sidelobes are widely used in many fields, such as communication and radar systems. Besides the correlation property, the frequency stopband property is often considered in the sequence design when the systems work in a crowded electromagnetic environment. In this paper, we aim at designing periodic/aperiodic sequences with low autocorrelation sidelobes and arbitrary frequency stopbands, and propose an efficient algorithm named FFT (fast Fourier transform)-based conjugate gradient algorithm. To calculate the step size efficiently, a method based on Taylor series expansion is developed. By changing the number of FFT points, the proposed algorithm can be easily used to generate periodic/aperiodic sequences. Since the gradient and step size can be implemented by FFT operations and Hadamard product, the whole algorithm is computationally efficient and can be used to design very long sequences. Numerical experiments show that the proposed algorithm has better performance than the state-of-the-art algorithms in terms of the running time.