Yaoran Yang

Yutong Zhang, Yaoran Yang

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.

Yutong Zhang, Yaoran Yang

This paper gives a unified algebraic solution to seven open problems of Wang, Tang and Ding on cyclic, negacyclic and constacyclic codes supporting designs. For the cyclic code \[ C\left(\frac{p^s-1}{2},\frac{p^s+1}{2}\right), \] a Cayley parametrization of the unit circle reduces the trace-zero condition to a semilinear equation on \(\PG(1,q)\). Its large root sets are exactly the \(\F_{p^{\gcd(m,s)}}\)-sublines, yielding the complementary design \[ \overline{S(3,q_0+1,q+1)}. \] For the length \(q^2+1\) negacyclic code, a quotient transport from \(\U_{2(q^2+1)}\) to \(\U_{q^2+1}\) and a unit-circle parametrization show that the minimum zero sets are precisely the Baer sublines of \(\PG(1,q^2)\). Equivalently, the corresponding support design is the complement of the non-tangent plane sections of an elliptic quadric \(\Q^-(3,q)\). For constacyclic ovoid codes of length \(q^2+1\) over \(\F_q\), the exact existence criterion is \[ λ\in\F_q^*,\qquad \exists\ λ\text{-constacyclic ovoid code} \Longleftrightarrow λ\notin(\F_q^*)^2. \] In particular, negacyclic ovoid codes exist exactly when \(q\equiv3\pmod4\). The proof uses the corrected projective-order congruence \[ a=(q+1)c,\qquad c\equiv b\pmod{q-1},\qquad \operatorname{ord}(θ\F_q^*)=\frac{q^2+1}{\gcd(q^2+1,c)}. \] The paper also derives a universal weight enumerator for lifted ovoid codes over extension fields, independent of the chosen ovoid. Finally, consecutive-root negacyclic MDS codes are constructed to give complete simple \(5\)-designs, including a proper negacyclic \([11,5,7]_{23}\) code whose minimum supports form the complete \(5-(11,7,15)\) design.

Yutong Zhang, Yaoran Yang, Yifan Zhu et al.

We solve the negative constant-curvature case of the COLT 2023 open problem of Criscitiello, Martínez-Rubio, and Boumal on deterministic first-order methods for Lipschitz geodesically convex optimization. Let \[ \HH^d_{-\kappaC^2}=\{X\in\R^{d+1}:\ipL{X}{X}=-1,\ X_0>0\}, \qquad \ip{U}{V}_{X}=\kappaC^{-2}\ipL{U}{V}, \] so the sectional curvature is $-\kappaC^2$. If \[ f:\bar B_{\HH}(x_0,r)\to\R \] is geodesically convex and $M$-Lipschitz, and $s=\kappaC r$, our one-shot Klein cutting-plane method returns a queried point $\hat x$ with \[ f(\hat x)-\min_{\bar B_{\HH}(x_0,r)}f\le \eps Mr \] using at most \[ \left\lceil 2d(d+1) \log\!\left(\frac{16\sinh s\cosh s}{s\eps}\right)\right\rceil \] oracle calls. For $d\ge2$ each localization update costs $O(d^2)$ arithmetic operations; for $d=1$ an interval variant satisfies the same bound. Consequently \[ N=O\bigl(d^2(s+\log(e/\eps))\bigr) =O\bigl(d^2ζ_s\log(e/\eps)\bigr), \qquad ζ_s=s/\tanh s . \] The argument is not a convex coordinate pullback: in the Beltrami--Klein chart the objective is generally only quasiconvex. The key point is that every Riemannian subgradient halfspace becomes an exact Euclidean central cut. For \[ θ=\kappaC\dist(X,Y), \] \[ \ip{g}{\log_XY}_{X} =\fracθ{\kappaC^2\sinhθ}\ipL{g}{Y}, \] and tangency at $X$ turns $\ipL{g}{Y}\le0$ into \[ \gbar^{\mathsf T}(u-c)\le0, \qquad u=Φ(Y),\quad c=Φ(X). \] Thus a fixed Euclidean ellipsoid localizes the whole hyperbolic ball. The only curvature payment is the Klein distortion factor \[ \log\left(\frac{\sinh s\cosh s}{s\eps}\right) =\log(1/\eps)+2s-\log(4s)+O(e^{-4s}). \]

Yaoran Yang, Yutong Zhang

A 2024 paper of Sun, Ding and Wang introduced a second class of constacyclic codes over finite fields, denoted $C(q,m,r,\ell)$, with length $(q^m-1)/r$, where $r\mid(q-1)$ and the defining monomials have total $q$-ary degree congruent to $r-1$ modulo $r$. In the non-projective intermediate range $2<r<q-1$ the paper gave a sharp-looking upper bound and a BCH-type lower bound, and left the minimum distance open. We prove that the upper bound is the exact minimum distance for every admissible intermediate parameter. More precisely, if $\ell=(q-1)a+b<(q-1)m-1$, $0\le b\le q-2$, and $b\equiv r-1\pmod r$, then, for every prime power $q$, every divisor $r$ of $q-1$ with $2<r<q-1$, and every $m\ge2$, \[ d(C(q,m,r,\ell))= \begin{cases} \displaystyle \frac{q-1}{r}(q-b+1)q^{m-a-2},&0\le a\le m-2,\\[1mm] \displaystyle \frac{q-b+r-2}{r},&a=m-1. \end{cases} \] The first line settles the open problem of Sun, Ding and Wang; the second line is the terminal case already forced by their BCH bound. We also determine the minimum affine support of every non-terminal scalar-residue layer of a generalized Reed--Muller code. The resulting dichotomy says that the first Reed--Muller weight survives exactly for residue classes $0$ and $1$, while every other residue-matched layer starts at the second Reed--Muller weight. The proof uses the hidden scalar homogeneity of the evaluation model, an orbit-counting obstruction for minimum Reed--Muller supports, and a homogeneous pencil construction that attains the second weight.