New Sequences Design From Weil Representation With Low Two-Dimensional Correlation in Both Time and Phase Shifts

A new elementary expression of the construction first proposed by Gurevich, Hadani, and Sochen is given, which avoids the explicit use of the Weil representation. The sequences in this signal set are given by both multiplicative character and additive character of finite fieldₚ. Such a signal set consists ofp²(p-2)time-shift distinct sequences, the magnitude of the two-dimensional autocorrelation function (i.e., the ambiguity function) in both time and phase of each sequence is upper bounded by2√pat any shift not equal to (0, 0). Furthermore, the magnitude of their Fourier transform spectrum is less than or equal to 2. For a subset consisting of$p(p-2)$phase-shift distinct sequences in this signal set, the magnitude of the ambiguity function of any pair is upper bounded by4√p. A proof is given through finding a new expression of the sequences in the finite harmonic oscillator system. An open problem for directly establishing these assertions without involving the Weil representation is addressed.