A Local Valuation Criterion for Quadratic-Permutation Interleaved Zadoff--Chu Sequences
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| Format: | Preprint |
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2026
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| _version_ | 1866916055169892352 |
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| author | Zhang, Yutong Yang, Yaoran |
| author_facet | Zhang, Yutong Yang, Yaoran |
| contents | Berggren and Popović introduced quadratic-permutation-polynomial interleaved Zadoff--Chu sequences and, from exhaustive data, conjectured that all normalized QPP-interleaved Zadoff--Chu sequences are inequivalent to ordinary Zadoff--Chu sequences precisely for prime-power lengths $N=p^n$ with $p>3$ and $n>1$. We give an exact local arithmetic criterion. For a normalized QPP $π_{a,b}(k)=ak^2+bk\pmod N$, the interleaved sequence is equivalent, under the standard five CAZAC-preserving operations, to a Zadoff--Chu sequence if and only if, for every prime power $p^α\Vert N$, the valuation of $a$ satisfies \[ ν_p(a)\ge \begin{cases} 0, & p=2,\ α=1,\\ α-1, & p=2,\ α\ge2,\\ α-1, & p=3,\\ α, & p>3. \end{cases} \] The proof is based on a third finite-difference invariant of the lifted Zadoff--Chu phase, namely \[ Δ^3\bigl((ak^2+bk+\varepsilon_N+2q)(ak^2+bk)\bigr)
=12a(2ak+3a+b). \] As a consequence, the conjectured prime-power boundary is not correct: the exact non-vacuous condition for all nonzero normalized QPPs to be inequivalent to Zadoff--Chu sequences is that $N$ is odd, $9\nmid N$, and $p^2\mid N$ for at least one prime $p\ge5$. In particular, $N=75=3\cdot5^2$ is the smallest non-prime-power counterexample to the conjectured ``only if'' direction. A second corollary records the corresponding statement for irreducible QPPs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20947 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Local Valuation Criterion for Quadratic-Permutation Interleaved Zadoff--Chu Sequences Zhang, Yutong Yang, Yaoran Number Theory Information Theory Berggren and Popović introduced quadratic-permutation-polynomial interleaved Zadoff--Chu sequences and, from exhaustive data, conjectured that all normalized QPP-interleaved Zadoff--Chu sequences are inequivalent to ordinary Zadoff--Chu sequences precisely for prime-power lengths $N=p^n$ with $p>3$ and $n>1$. We give an exact local arithmetic criterion. For a normalized QPP $π_{a,b}(k)=ak^2+bk\pmod N$, the interleaved sequence is equivalent, under the standard five CAZAC-preserving operations, to a Zadoff--Chu sequence if and only if, for every prime power $p^α\Vert N$, the valuation of $a$ satisfies \[ ν_p(a)\ge \begin{cases} 0, & p=2,\ α=1,\\ α-1, & p=2,\ α\ge2,\\ α-1, & p=3,\\ α, & p>3. \end{cases} \] The proof is based on a third finite-difference invariant of the lifted Zadoff--Chu phase, namely \[ Δ^3\bigl((ak^2+bk+\varepsilon_N+2q)(ak^2+bk)\bigr) =12a(2ak+3a+b). \] As a consequence, the conjectured prime-power boundary is not correct: the exact non-vacuous condition for all nonzero normalized QPPs to be inequivalent to Zadoff--Chu sequences is that $N$ is odd, $9\nmid N$, and $p^2\mid N$ for at least one prime $p\ge5$. In particular, $N=75=3\cdot5^2$ is the smallest non-prime-power counterexample to the conjectured ``only if'' direction. A second corollary records the corresponding statement for irreducible QPPs. |
| title | A Local Valuation Criterion for Quadratic-Permutation Interleaved Zadoff--Chu Sequences |
| topic | Number Theory Information Theory |
| url | https://arxiv.org/abs/2605.20947 |