This paper introduces a novel type of sequences called C4-sequences. C4-sequences share similar optimal autocorrelation properties with Zadoff-Chu sequences. However, C4-sequences offer the additional advantage of being also optimal (in the sense of minimal Euclidean distance between sequences) for four truncation lengths, providing flexibility in adapting to different channel conditions without compromising performance. Moreover, unlike Zadoff-Chu sequences, the points of a constellation associated with a C4-sequence are not limited to the unit circle. This opens up possibilities for achieving shaping gain, leading to enhanced spectral efficiency. By combining a truncated C4-sequence modulation as an inner code with a fixed-rate non-binary outer code, flexible and performant rate-adaptive communication systems can also be achieved. Finally, the notion of C4-sequences can be generalized.