Low Ambiguity Zone (LAZ) sequences are essential in modern integrated sensing and communication (ISAC) systems. Recently, locally perfect nonlinear functions (LPNFs) have been employed to design LAZ sequences with flexible parameters. In this work, we propose three new classes of LPNFs and use them to construct LAZ sequences that offer additional flexible parameters. Notably, all the proposed LAZ sequences are asymptotically optimal with respect to the Ye-Zhou-Fan-Liu-Lei-Tang bounds derived in [IEEE J. Sel. Areas Commun. 40(6), pp. 1809-1822, 2022].

In the most interesting case of safe prime powers q, Gómez and Winterhof showed that a subfamily of the family of Golomb Costas permutations of$\{1,2,\ldots,q-2\}$of size$\varphi (q-1)$has maximal cross-correlation of order of magnitude at most$q^{1/2}$. In this paper we study a larger family of Golomb Costas permutations and prove a weaker bound on its maximal cross-correlation. Considering the whole family of Golomb Costas permutations we show that large cross-correlations are very rare. Finally, we collect several conditions for a small cross-correlation of two Costas permutations. Our main tools are the Weil bound and the Szemerédi-Trotter theorem for finite fields.