Permutation Polynomial CAZAC Constructions

• Permutation polynomial-based CAZAC constructions are sequence design methods that use high-degree permutation polynomials to generate constant amplitude sequences with zero off-peak autocorrelation.
• The methodology interleaves classical Zadoff–Chu sequences with algebraic permutations to rigorously achieve orthogonality and optimal correlation properties using group-theoretic and exponential-sum techniques.
• These constructions offer practical benefits in radar, wireless, and multicarrier systems by enabling flexible sequence allocation, efficient synchronization, and reduced interference with low computational overhead.

Permutation polynomial-based CAZAC constructions form a contemporary branch of sequence design for radar, wireless communications, and signal processing, leveraging algebraic permutations for generating constant amplitude zero autocorrelation (CAZAC) sequences. These constructions generalize and extend the classic Zadoff–Chu (ZC) sequence approach by employing high-degree permutation polynomials (PPs) and their properties over integer rings or finite fields, leading to new sequence families with rigorous correlation and orthogonality properties and substantial impact in both theoretical and applied domains (Yuan et al., 17 Jan 2026, Berggren et al., 2023, Wang et al., 2020, Wang et al., 2018).

1. Mathematical Foundations: Ring Structures, Zadoff–Chu, and Permutation Polynomials

Let $N \in \mathbb{N}$, and define the residue ring $\mathbb{Z}_N = \{0,1,\dots,N-1\}$, with operations modulo $N$. Given a root index $u$ coprime to $N$, the canonical ZC sequence is

$$s(k) = e^{-2\pi i u [k^2 + (N \bmod 2) k]/(2N)},\quad 0 \leq k < N.$$

This sequence satisfies $|s(k)|=1$ and possesses ideal (zero out-of-phase) periodic autocorrelation: for $0 < d < N$, $\theta_s(d) = \sum_{k=0}^{N-1} s(k) s^*(k+d) = 0$.

A polynomial $P(x) \in \mathbb{Z}_N[x]$ is a permutation polynomial (PP) if $\mathbb{Z}_N = \{0,1,\dots,N-1\}$0 permutes $\mathbb{Z}_N = \{0,1,\dots,N-1\}$1. For prime power or $\mathbb{Z}_N = \{0,1,\dots,N-1\}$2-power $\mathbb{Z}_N = \{0,1,\dots,N-1\}$3, explicit coefficient criteria determine when a given $\mathbb{Z}_N = \{0,1,\dots,N-1\}$4 is a PP (e.g., Rivest’s conditions for $\mathbb{Z}_N = \{0,1,\dots,N-1\}$5). A significant class is $\mathbb{Z}_N = \{0,1,\dots,N-1\}$6, for suitable $\mathbb{Z}_N = \{0,1,\dots,N-1\}$7, and $\mathbb{Z}_N = \{0,1,\dots,N-1\}$8 having a specific prime-power factorization and additional congruence conditions.

The compositional inverse $\mathbb{Z}_N = \{0,1,\dots,N-1\}$9 is itself a PP, although explicit inversion formulas exist only for low-degree cases.

2. Construction: Interleaving ZC Sequences with Permutation Polynomials

Given a sequence $N$0 of length $N$1 and a PP $N$2 on $N$3, define the interleaved sequence $N$4 via $N$5. If $N$6 is a ZC sequence, $N$7 retains constant amplitude, and its periodic autocorrelation is

$N$8

For quadratic permutation polynomials (QPPs) and higher-degree examples from the prime-factor class, the exponent’s algebraic structure allows a group-theoretic or exponential-sum approach to proving $N$9 for all $u$0 (Yuan et al., 17 Jan 2026, Berggren et al., 2023).

A significant extension is that for any PP $u$1 whose inverse is quadratic or linear, $u$2 remains CAZAC (Berggren et al., 2023). Invariably, the operation $u$3 produces new CAZAC sequences outside the orbit of the classical ZC and linear FM classes.

3. Correlation Properties, CAZAC Proofs, and Aperiodic Performance

The central feature is the CAZAC property: $u$4 and $u$5 for $u$6. For high-degree PPs, the periodic autocorrelation is handled by analyzing the exponent polynomial's shift properties. For the prime-factor class, one can select a translation step $u$7 such that $u$8 is constant but nonzero modulo $u$9, ensuring exponential sum cancellation by nontrivial rotation (Yuan et al., 17 Jan 2026).

For $N$0, the Rivest criterion on PP coefficients and considerations on induced quadratic forms in the autocorrelation sum ensure vanishing off-peak autocorrelation by difference-of-squares and permutation arguments.

Aperiodic autocorrelation is increasingly relevant in communication systems using non-circular convolutions. For QPP interleavers of ZC over $N$1, the maximum aperiodic correlation magnitude satisfies $N$2 for $N$3 and some constant $N$4 independent of $N$5 (Yuan et al., 17 Jan 2026). The proof applies Weyl sum methods and combinatorial bounds on exponential sums.

4. Equivalence Classes and Enumeration of Novelty

CAZAC sequences are considered up to the five basic CAZAC-preserving operations: rotation, translation, decimation, linear FM, and complex conjugation. The equivalence class of a sequence is its orbit under these operations (Yuan et al., 17 Jan 2026, Berggren et al., 2023).

It is established that the new classes constructed via high-degree PPs are inequivalent to classic ZC and QPP-interleaved sequences. The proof involves demonstrating that the necessary functional identity between exponents would yield vanishing coefficients for high-degree monomials, typically via a Vandermonde-type argument, leading to contradiction unless the PP is of lower degree (Yuan et al., 17 Jan 2026).

5. Families, Orthogonality, and Practical Design

Permutation-polynomial interleaving greatly expands the CAZAC sequence family, generating inequivalent sets often unavailable via classical approaches. For select QPPs over $N$6, it is possible to construct mutually orthogonal CAZAC families, enabling user or cell-specific allocation for interference avoidance (Berggren et al., 2023). Orthogonality arises when the auxiliary modulations induced by QPPs form orthogonal sets under the discrete inner product, with explicit construction methods for binary and quaternary cases.

This methodology also finds deep connections to complementary sequence sets (CSS), complete complementary codes (CCC), and field-trace-based constructions in the finite field setting (Wang et al., 2020, Wang et al., 2018). In these frameworks, PPs, often combined with field traces, seed matrices (e.g., Butson–Hadamard) whose rows generate CAZAC and complementary sequence sets, ensuring zero aperiodic and periodic cross-correlation across families.

6. Algebraic Structures, Extensions, and Generalizations

Beyond $N$7, permutation polynomials over finite fields $N$8 allow the construction of CAZAC sequences of length $N$9 via $$s(k) = e^{-2\pi i u [k^2 + (N \bmod 2) k]/(2N)},\quad 0 \leq k < N.$$0, where $$s(k) = e^{-2\pi i u [k^2 + (N \bmod 2) k]/(2N)},\quad 0 \leq k < N.$$1 is a PP and $$s(k) = e^{-2\pi i u [k^2 + (N \bmod 2) k]/(2N)},\quad 0 \leq k < N.$$2 is the field trace (Wang et al., 2018). Multivariate reduction and resultant elimination techniques identify large classes of sparse and high-degree permutation polynomials suitable for sequence design.

The algebraic-combinatorial framework developed by Wang and Gong connects these constructions with Reed–Muller and Golay codes, allowing the joint optimization of correlation and error-correcting properties, and providing flexible codebooks for multicarrier (OFDM) systems with bounded PMEPR (Wang et al., 2020).

7. Impact, Implementation Considerations, and Outlook

The expansion of CAZAC families by permutation polynomial interleaving has direct impacts on standards such as 3GPP 38.211, where flexibility in generating orthogonal or user-specific sequences is crucial for synchronization, random access, and reference signaling. Implementation is efficient, since evaluation of PPs and their inverses modulo $$s(k) = e^{-2\pi i u [k^2 + (N \bmod 2) k]/(2N)},\quad 0 \leq k < N.$$3 can be realized with low computational overhead (Berggren et al., 2023).

The proven sufficiency of the Berggren–Popović conjecture, namely, that QPP interleaving for $$s(k) = e^{-2\pi i u [k^2 + (N \bmod 2) k]/(2N)},\quad 0 \leq k < N.$$4, $$s(k) = e^{-2\pi i u [k^2 + (N \bmod 2) k]/(2N)},\quad 0 \leq k < N.$$5, produces sequences inequivalent to standard ZC unless the PP is linear, lends theoretical completeness to the classification of permutation-interleaved CAZACs (Yuan et al., 17 Jan 2026).

A plausible implication is that further investigation into higher-degree PPs, especially those not covered by current classification theorems, may yield additional CAZAC sequence types and richer structures, possibly with specialized correlation or PMEPR properties for advanced communication and sensing scenarios.