Character Sums and Polyphase Sequence Families with Low Correlation, DFT and Ambiguity

We present a survey on the current status of the constructions of polyphase sequences with low correlation, discrete Fourier transform (DFT), and ambiguity in both time and phase domain, including some new insights and results. Firstly, we systematically introduce the concepts of phase-shift operators and ambiguity functions of sequences, and give a new construction of polyphase sequences from combinations of different indexing field elements and hybrid characters. We then present the constructions, some known and some new, of polyphase sequences with low degree polynomials, for their low correlation, DFT and ambiguity can be bounded by directly applying the Weil bounds. Thirdly, we introduce the Hadamard equivalence, restate the conjectured new ternary 2-level autocorrelation sequences, and present their Hadamard equivalence relations. Some open problems are presented.