Channel estimation in time division duplexing (TDD)-based massive multiple-input multiple-output (MIMO) systems is heavily hampered by the pilot contamination, which constitutes a major bottleneck on the overall system performance. This study considers the pilot contamination problem in multi-cell TDD-based massive MIMO systems, and analytical expressions are presented on the normalised mean square error (NMSE) of the minimum mean square error channel estimation algorithm. Based on the obtained NMSE, this study proposes an optimal pilot assignment strategy to minimise the effect of pilot contamination. In order to further improve the system performance, a pilot design-based channel estimation scheme is proposed, where Chu sequences with perfect auto-correlation property are employed to design the optimal pilot sequences aiming at acquiring the accurate channel state information. Simulation results show that the proposed pilot assignment strategy outperforms the random pilot assignment method, and approaches to the performance of the exhaustive search method which requires high computational complexity. Moreover, the performance gain of the pilot design-based channel estimation scheme is verified in massive MIMO systems.