[Paper Review] An Explicit Upper Bound of Generalized Quadratic Gauss Sums and Its Applications for Asymptotically Optimal Aperiodic Polyphase Sequence Design

Huaning Liu, Zilong Liu|arXiv (Cornell University)|Jan 23, 2026

Quasicrystal Structures and Properties0 citations

TL;DR

The paper proves an explicit upper bound for generalized quadratic Gauss sums using Paris’ asymptotic expansion and Fibonacci zeta convergence, then uses this bound to construct four families of order-optimal aperiodic polyphase sequence sets based on Chu and Alltop sequences.

ABSTRACT

This work is motivated by the long-standing open problem of designing asymptotically order-optimal aperiodic polyphase sequence sets with respect to the celebrated Welch bound. Attempts were made by Mow over 30 years ago, but a comprehensive understanding to this problem is lacking. Our first key contribution is an explicit upper bound of generalized quadratic Gauss sums which is obtained by recursively applying Paris' asymptotic expansion and then bounding it by leveraging the fast convergence property of the Fibonacci zeta function. Building upon this major finding, our second key contribution includes four systematic constructions of order-optimal sequence sets with low aperiodic correlation and/or ambiguity properties via carefully selected Chu sequences and Alltop sequences. For the first time in the literature, we reveal that the full Alltop sequence set is asymptotically optimal for its low aperiodic correlation sidelobes. Besides, we introduce a novel subset of Alltop sequences possessing both order-optimal aperiodic correlation and ambiguity properties for the entire time-shift window.

Motivation & Objective

• Motivate the long-standing problem of designing asymptotically order-optimal aperiodic polyphase sequence sets with respect to the Welch bound.
• Derive an explicit upper bound for generalized quadratic Gauss sums to enable tight performance guarantees for aperiodic sequences.
• Develop systematic constructions of order-optimal sequence sets with low aperiodic correlation and/or ambiguity properties.
• Show that Alltop sequences (full set) are asymptotically optimal for low aperiodic sidelobes and provide a subset achieving simultaneous order-optimality for correlation and ambiguity.

Proposed method

• Apply Paris’ asymptotic expansion recursively to S_N(x, theta) and bound the resulting error terms.
• Introduce a scenario-specific error control for the complementary error function E(x, theta) to obtain an explicit bound.
• Exploit Fibonacci zeta function convergence to bound the accumulation of error terms and derive a numerical bound |S_N(a/q, theta)| < 20.07 sqrt(q) + 3.
• Construct four sequence families using Chu and Alltop sequences: two Chu-based families (C1, C2) and two Alltop-based families (A1, A2).
• Prove C1 is an order-optimal low-correlation set (delta_max <= max{21 sqrt(L), 0.35 sqrt(KL)}).
• Prove C2 provides an order-optimal set with aperiodic LAZ properties and bound theta_max; show A1 is asymptotically optimal for low aperiodic sidelobes and A2 achieves simultaneous asymptotic optimality for correlation and ambiguity.

Experimental results

Research questions

• RQ1 Can an explicit, computable upper bound be obtained for generalized quadratic Gauss sums to guide sequence design?
• RQ2 Can explicit bounds enable constructive, asymptotically order-optimal aperiodic polyphase sequence sets based on Chu and Alltop sequences?
• RQ3 What are the correlation and ambiguity performance guarantees (asymptotically) for the proposed Chu- and Alltop-based sequence families?

Key findings

• An explicit bound is proven: |S_N(a/q, theta)| < 20.07 sqrt(q) + 3 for q ≥ 2, N ≤ q, gcd(a,q)=1.
• Two Chu-based constructions (C1 and C2) and two Alltop-based constructions (A1 and A2) yield four order-optimal sequence families with controlled aperiodic correlation/ambiguity.
• C1 yields low aperiodic correlation with delta_max(C1) ≤ max{21 sqrt(L), 0.35 sqrt(KL)} and includes Mow’s Chu pair as a special case (K=2).
• C2 delivers aperiodic LAZ guarantees with theta_max(C2) ≤ (1.35 + 2.035/√m) sqrt(L) + 5 under specified parameters.
• Alltop-based A1 set is shown to be asymptotically optimal for its low aperiodic sidelobes, and its subset A2 achieves simultaneous asymptotic optimality in both correlation and ambiguity over the full time-shift window.