This paper studies Flag sequences for lowcomplexity delay-Doppler estimation by exploiting their distinctive peak-curtain ambiguity functions (AFs). Unlike the existing Flag sequence designs that are limited to prime lengths and periodic auto-AFs, we aim to design Flag sequence sets of arbitrary lengths and with low (nontrivial) periodic/aperiodic auto- and cross-AFs. Since every Flag sequence consists of a Curtain sequence and a Peak sequence, we first investigate the algebraic design of zone-based Curtain sequence sets of arbitrary lengths. Our proposed design gives rise to novel Curtain sequence sets with ideal curtain auto-AFs and low/zero cross-AFs within the delay-Doppler zone of interest. Leveraging these Curtain sequence sets, two optimization problems are formulated to minimize the summed customized weighted integrated sidelobe level (SCWISL) of the Flag sequence set. Accelerated Parallel Partially Majorization-Minimization Algorithms are proposed to jointly optimize the transmit Flag sequences and matched/mismatched reference sequences stored in the receiver. Simulations demonstrate that our proposed Flag sequences lead to improved SCWISL and customized peak-to-max-sidelobe ratio compared with the existing Flag sequences. Additionally, our Flag sequences under Flag method exhibit Mean Squared Errors that approach the Cramer-Rao Lower Bound and the Sampling Bound at high signal-to-noise power ratios.

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