<p>Sequences play an important role in communication and radar systems, where related theoretical bounds serve as benchmarks to access the designed sequences. This paper first studies the odd-periodic ambiguity function (OPAF) of sequence sets and derives a theoretical lower bound of its magnitude. Based on the even-odd transformation, the relationship between the periodic ambiguity function and the odd-periodic ambiguity function is established. Furthermore, we construct two classes of odd-periodic sequences that achieve the optimal zero ambiguity zone (ZAZ) by leveraging the properties of known sequences and quadratic functions. Finally, we present a class of low ambiguity zone (LAZ) sequence sets by using certain cubic functions, which is asymptotically optimal with respect to the derived bound.</p>