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Harmonic higher weight distributions, Simonis' approach of MacWilliams identity and moments

We present a combinatorial proof of Simonis type MacWilliams identity for harmonic higher weight distributions of linear codes. Furthermore, we investigate the statistical moments of the harmonic higher weight enumerators for random linear codes. Defining the enumerators via rank functions of the generator matrices of linear codes, we prove that its expectation vanishes for all non-trivial harmonic functions due to the inherent symmetry of random matrices, and we also derive an explicit, non-trivial formula for the covariance.

math.CO

New classes of trace-form permutation polynomials and their compositional inverses

We construct permutation polynomials of the form x + γTr^{q^2}_q(h(x)) over the finite field F_{q^2}. More precisely, we present two families ofpermutation polynomials whose coefficients range over all elements of the field rather than being restricted to a proper subfield. We also compute the compositional inverses of both families. Our techniques involve the evaluation of certain Kloosterman sums together with the analysis of some equations over finite fields, and we believe these can be of independent interest.

math.NT

Permutation polynomials over finite fields from low-degree rational functions

This paper considers permutation polynomials over the finite field $F_{q^2}$ in even characteristic by utilizing low-degree permutation rational functions over $F_q$. As a result, we obtain two classes of permutation binomials and six classes of permutation pentanomials over $F_{q^2}$. Additionally, we show that the obtained binomials and pentanomials are quasi-multiplicative inequivalent to the known ones in the literature.

cs.CR

Deciding superellipticity and computing the Weierstrass normal form

Let \( \mathcal{S}_{g,n} \subset \mathcal{M}_g \) be the locus of curves of genus \( g \geq 2 \) admitting a model \( y^n = h(x) \) with \( h \) separable; such curves $C$ have a cyclic group \( C_n \leq \operatorname{Aut}(C) \) of order \( n \) with \( C/C_n \cong \mathbb{P}^1 \). % We give an algorithm which, given an absolutely irreducible plane model \( F(x,y) = 0 \) of a curve \( C \) over a field \( k_0 \) of characteristic zero, decides for which \( n \) the curve lies in \( \mathcal{S}_{g,n} \) and returns a model \( y^n = h(x) \) together with the birational transformation to it.

math.AG

Differential uniformity properties of some classes of permutation polynomials

The notion of $c$-differential uniformity has recently received a lot of attention since its proposal~\cite{Ellingsen}, and recently a characterization of perfect $c$-nonlinear functions in terms of difference sets in some quasigroups was obtained in~\cite{AMS22}. Independent of their applications as a measure for certain statistical biases, the construction of functions, especially permutations, with low $c$-differential uniformity is an interesting mathematical problem in this area, and recent work has focused heavily in this direction. We provide a few classes of permutation polynomials with low $c$-differential uniformity. The used technique involves handling various Weil sums, as well as analyzing some equations in finite fields, and we believe these can be of independent interest.

cs.IT

Eleven, twelve, and thirteen lonely runners

Wills conjectured that, for any non-zero integers $u_1,\ldots,u_k$, there is a real number $t$ such that, for all $i=1,\ldots,k$, \[\lVert tu_i\rVert\geq\frac{1}{k+1},\] where $\lVert x\rVert$ is the distance from $x$ to the closest integer. This statement is known as the Lonely Runner Conjecture. A computational method developed by Rosenfeld and the second author verified the conjecture for $k\leq9$. We further refine this method with new sieving techniques and employ a polynomial method argument to show that any $(u_1,\ldots,u_k)\equiv(1,2,\ldots,k)\pmod{p}$ with $\gcd(u_1,\ldots,u_k)=1$ satisfies the conjecture when $k+1$ and $p > k^2+k$ are both odd primes. Ultimately, we provide a computer-assisted proof of the Lonely Runner Conjecture for $k\in\{10,11,12\}$.

math.CO

The second-order zero differential spectra of some APN and other maps over finite fields

The Feistel Boomerang Connectivity Table and the related notion of $F$-Boomerang uniformity (also known as the second-order zero differential uniformity) has been recently introduced by Boukerrou et al.~\cite{Bouk}. These tools shall provide a major impetus in the analysis of the security of the Feistel network-based ciphers. In the same paper, a characterization of almost perfect nonlinear functions (APN) over fields of even characteristic in terms of second-order zero differential uniformity was also given. Here, we find a sufficient condition for an odd or even function over fields of odd characteristic to be an APN function, in terms of second-order zero differential uniformity. Moreover, we compute the second-order zero differential spectra of several APN or other low differential uniform functions, and show that our considered functions also have low second-order zero differential uniformity, though it may vary widely, unlike the case for even characteristic when it is always zero.

cs.IT

A General Construction of Codes from Drinfeld Modules

We construct additive rank-metric and sum-rank-metric codes from Drinfeld modules by restricting bounded-degree morphisms to prime-to-characteristic torsion. For supersingular Drinfeld modules of rank $r$ in characteristic $\mathfrak{p}$ of degree $d$, the stabilization formula for morphism spaces yields rank-metric codes of $\mathbb{F}_q$-dimension $mrt-c$ and minimum distance $r-t+1$, where $c=r(r-1)(d-1)/2$. Simultaneous restriction to $\ell$ distinct degree-$m$ torsion modules gives additive sum-rank codes of the same dimension and minimum distance at least $\ell r-t+1$. Their normalized Singleton defects tend to zero, while in characteristic $(T)$ the module $ϕ_T=τ^r$ makes the defect vanish and produces an explicit MSRD family. We identify this family with a skew Chinese remainder theorem code supported on central skew polynomials and prove that its poly-skew weight is exactly $m$ times its sum-rank weight. This gives a specialized Singleton-type bound and a polynomial-time unique decoder up to the full sum-rank unique-decoding radius. We also derive a Welch-Berlekamp-type filter equation for the general supersingular sum-rank construction; it becomes an effective decoder whenever bases of the relevant morphism spaces and the restriction maps are computable.

math.NT

The second-order zero differential spectra of some functions over finite fields

It was shown by Boukerrou et al.~[IACR Trans. Symmetric Cryptol. 1 (2020), 331--362] that the $F$-boomerang uniformity (which is the same as the second-order zero differential uniformity in even characteristic) of perfect nonlinear functions is~$0$ on $\F_{p^n}$ ($p$ prime) and the one of almost perfect nonlinear functions on $\F_{2^n}$ is~$0$. It is natural to inquire what happens with APN or other low differential uniform functions in even and odd characteristics. Here, we explicitly determine the second-order zero differential spectra of several maps with low differential uniformity. In particular, we compute the second-order zero differential spectra for some almost perfect nonlinear (APN) functions over finite fields of odd characteristic, pushing further the study started in Boukerrou et al. and continued in Li et al.~[Cryptogr. Commun. 14.3 (2022), 653--662], and it turns out that our considered functions also have low second-order zero differential uniformity. Moreover, we study the second-order zero differential spectra of certain functions with low differential uniformity over finite fields of even characteristic. We connect this new concept to the sum-freedom and vanishing flats concepts and find some counts for the number of vanishing flats via our methods. We provide detailed analyses on several equations over finite fields that may have an interest outside of the scope of our paper.

cs.IT

Logarithmic Chowla Correlations Across All Shift Scales

Let $λ(n)=(-1)^{Ω(n)}$ be the Liouville function. We prove a fixed power-logarithmic bound for its logarithmically weighted two-point correlations across the full shift range. There is an absolute $c>0$ such that every sufficiently large $x$ admits a single set $\mathcal E_x\subseteq[1,x]$ with $|\mathcal E_x\cap[1,H]|\ll_A H(\log x)^{-A}$ $(1\le H\le x)$ for every fixed $A>0$, while $\max_{\substack{1\le h\le x\ h\notin\mathcal E_x}}\sup_{1\le y\le x}\left|\sum_{n\le y}\frac{λ(n)λ(n+h)}{n}\right|\ll(\log x)^{1-c}$. The same exceptional-set formulation extends, without an upper cutoff, to all positive integer shifts. Earlier full-range theorems average over the shift; here a fixed saving holds pointwise outside one set whose density in every initial segment is smaller than every fixed negative power of $\log x$. The new middle-scale argument combines a general-good-modulus Liouville deletion lemma with a linear bad-modulus score, a progression Fourier estimate, and a Mellin-localized dilation that separates divisor-dependent endpoints. Maximal fixed-moment bounds evacuate the low prefix and control the long-shift range. Assuming GRH for primitive Dirichlet $L$-functions, we also prove, uniformly for $h\in\mathbb N$ and $1\le y\le x$, $\left|\sum_{n\le y}\frac{λ(n)λ(n+h)}{n}\right|\le\log(2\min{h,y})+O((\log x)^{1-c_{\mathrm G}})$ for an absolute $c_{\mathrm G}>0$, with no exceptional shifts.

math.NT

Reduced polynomial lifts of APN permutations over Galois rings and effective non-APN bounds

We clarify the APN lifting conjecture over Galois rings of Rønjom and Sandrib (CCDS, 2026). A function on $\F_q$ has many polynomial representatives, whose formal derivatives may differ, so the conjecture must use the unique reduced representative of degree less than $q$; without this normalization, it is false. The standard permutation-polynomial criterion over Galois rings then gives an exact reduction: the reduced representative $f$ of an APN permutation lifts to a permutation of $\GR(2^k,m)$, $k>1$, if and only if $f'(x)\ne0$ for every $x\in\F_{2^m}$. Thus the corrected lifting conjecture is equivalent to a finite-field critical-point conjecture. We next use Janwa--Wilson--Rodier surfaces, which encode the APN condition by rational points off the diagonal arrangement, to prove an effective nonexistence result. For every odd degree $d\ge5$ outside the Gold exponents $2^r+1$ and Kasami--Welch exponents $2^{2r}-2^r+1$, results of Hernando--McGuire and Aubry--McGuire--Rodier provide an absolutely irreducible factor in the hyperplane section at infinity. This yields an absolutely irreducible component of the surface, defined over the ground field and not contained in the diagonal arrangement. The explicit Cafure--Matera estimate then gives a computable number $\APNmzero{d}$ such that no polynomial of degree $d$ over $\F_{2^m}$ is APN when $m\ge\APNmzero{d}$. The qualitative eventual non-APN result is due to Aubry--McGuire--Rodier; our contribution is the explicit threshold. A direct identity for difference tables also gives an even-degree consequence: if $g$ has such an odd degree, then $ax+g(x^2)+c$, with $a\ne0$, has the same differential uniformity as $g$ and is therefore not APN in the same explicit range. Finally, we prove directly that every cubic permutation polynomial has a rational critical point and hence satisfies the corrected lifting conjecture.

math.NT

Semi-Implicit Central scheme for Hyperbolic Systems of Balance Laws with Relaxed Source Term

Quasi-linear hyperbolic systems with source terms introduce significant computational challenges due to the presence of a stiff source term. To address this, a finite volume Nessyahu-Tadmor (NT) central numerical scheme is explored and applied to benchmark models such as the Jin-Xin relaxation model, the shallow-water model, the Broadwell model, the Euler equations with heat transfer, and the Euler system with stiff friction to assess their effectiveness. The core part of this numerical scheme lies in developing a new implicit-explicit (IMEX) scheme, where the stiff source term is handled in a semi-implicit manner constructed by combining the midpoint rule in space, the trapezoidal rule in time with a backward semi-implicit Taylor expansion. The advantage of the proposed method lies in its stability region and maintains robustness near stiffness and discontinuities, while asymptotically preserving second-order accuracy. The numerical validation further extends to two-dimensional configurations of the Jin-Xin relaxation model and a Jin-Xin-type relaxation system of 2D Euler equation. Theoretical analysis and numerical validation confirm the stability and accuracy of the method, highlighting its potential for efficiently solving the stiff hyperbolic systems of balance laws of 1D and 2D.

math.NA

Beyond sensitivity: mechanism-resolved error budgets for designing quantum sensors

Quantum sensors are specified by a headline sensitivity, yet applications also demand accuracy and reliability. The dominant limiter of one metric is often known, but no method resolves how interacting mechanisms combine into a signed, per-mechanism budget for each metric. We introduce a framework that computes a sensor's sensitivity, accuracy, and robustness from one open-system simulation and attributes each to its limiting mechanism. For a nitrogen-vacancy diamond ensemble the attribution inverts across metrics: dephasing limits sensitivity, the thermal ground-state shift limits accuracy, and optical leakage limits robustness. At identical sensitivity the recovered-field bias spans $8$ to $1500$\,nT, so tuning to sensitivity alone can miss the accuracy target by two orders of magnitude. The same modeling transfers to a cesium optically pumped magnetometer recording a human magnetocardiogram. As a digital twin, it predicts the gain from addressing each limiter, so sensors can be designed to the required metrics.

quant-ph

RIBOSPAN: A Long-Context RNA Foundation Model for Versatile RNA Modeling

Full-length RNAs, particularly messenger RNAs, often exceed the context lengths used to pretrain existing RNA foundation models, limiting complete-transcript modeling at single-nucleotide resolution. We present RIBOSPAN, a 1.61-billion-parameter bidirectional RNA foundation model natively pretrained with context lengths up to 10,240 nt. RIBOSPAN combines dense bidirectional self-attention, single-nucleotide tokenization, and attention-isolated sequence packing to enable high-resolution modeling of complete long RNAs. Native 10K pretraining preserves strong reconstruction at 10,240 tokens and, in a controlled long-context benchmark, maintains strong contextual responsiveness and context-specific representation separation while keeping perturbation-induced changes highly localized. Inference-time YaRN scaling recovers much of the contextual organization lost by direct short-context extrapolation, but induces substantially greater distal representation diffusion. Frozen RNA-type evaluations show that RIBOSPAN learns state-of-the-art RNA representations, with a particularly clear advantage on long RNAs. Across downstream biological benchmarks, RIBOSPAN emerges as the strongest encoder-only RNA foundation model, achieving state-of-the-art performance in both full-transcript biological property prediction and zero-shot mutation-fitness modeling. Building on the same backbone, we develop a multidimensionally conditioned discrete-diffusion framework for full-length mRNA generation and redesign, including synonymous-codon diffusion for protein-preserving CDS optimization. Together, RIBOSPAN establishes a powerful long-context foundation for transferable RNA representation learning, biological prediction, and full-transcript mRNA design.

cs.LG

Turing complete Navier-Stokes steady states via cosymplectic geometry

In this article, we construct stationary solutions to the Navier-Stokes equations on certain Riemannian $3$-manifolds that exhibit Turing completeness, in the sense that they are capable of performing universal computation. This universality arises on manifolds admitting nonvanishing harmonic 1-forms, thus showing that computational universality is not obstructed by viscosity, provided the underlying geometry satisfies a mild cohomological condition. The proof makes use of a correspondence between nonvanishing harmonic $1$-forms and cosymplectic geometry, which extends the classical correspondence between Beltrami fields and Reeb flows on contact manifolds.

math.DG

Geometric mean and Lebesgue-type decomposition of completely positive maps

We introduce the geometric mean and the parallel sum of completely positive (CP) maps between von Neumann algebras, based on the Pusz--Woronowicz theory of positive sesquilinear forms. We provide a concrete characterization via a block matrix positivity condition and establish their fundamental properties, including the AM--GM--HM inequality with respect to the CP order. In finite-dimensional settings, our construction is compatible with the Choi--Jamiolkowski correspondence, under which the geometric mean of CP maps corresponds to the Kubo--Ando geometric mean of their Choi matrices. This yields a natural operator-theoretic framework for interpolating quantum channels. As an application, we obtain index-type inequalities for conditional expectations in subfactor theory. Finally, we establish a Lebesgue-type decomposition of CP maps via a parallel sum construction, thereby providing a unified framework that simultaneously generalizes Ando's decomposition of bounded positive operators and Kosaki's decomposition of normal positive functionals on von Neumann algebras.

math.OA

Two Adjoint Perspectives on Fokker-Planck Optimization: A Microscopic-Macroscopic Correspondence

The Fokker-Planck equation admits both a macroscopic Eulerian description through probability densities and a microscopic Lagrangian description through stochastic trajectories. Consequently, optimization problems constrained by the Fokker-Planck equation can be formulated from either perspective. Surprisingly, the corresponding adjoint equations appear to be fundamentally different: the macroscopic adjoint is governed by the backward Kolmogorov equation, whereas the microscopic adjoint evolves pathwise along stochastic trajectories. In this note, we reconcile these two formulations by establishing their correspondence in the continuum setting. We further show that, although their discrete gradients no longer coincide after discretization, both provide consistent numerical approximations of the continuum gradient. Explicit convergence rates are established for both discretization strategies.

math.NA

Multiplicative comparisons of Rényi entropies for weighted Bernoulli sums

We establish improved multiplicative bounds relating the Rényi entropies of different orders for weighted sums of independent Bernoulli random variables. In particular, we prove a logarithmic bound between the zeroth-order and infinity-order Rényi entropies, which yields a polynomial improvement over the square-root bound of Jain, Sah, and Sawhney. Additionally, we obtain explicit constant-factor bounds for comparisons among Rényi entropies of nonzero orders.

math.PR