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arXiv2026

ISAC with Co-Prime Arrays: Virtual-Aperture Sensing and uplink downlink communications

Integrated sensing and communication (ISAC) enables simultaneous communication and environmental sensing in unmanned aerial vehicle (UAV) networks, but its performance is constrained by the physical antenna aperture and residual self-interference (SI) in full-duplex (FD) sensing. To address these issues, we propose a shared-aperture ISAC architecture in which a sparse co-prime array (CPA) is embedded in a uniform linear array (ULA) grid for FD sensing, while the remaining antenna positions support time-division duplexing (TDD) communication. We characterize the sensing performance through an order-wise Cramer-Rao bound (CRB) analysis, showing that the CPA achieves a stronger asymptotic sensing gain than the partitioned ULA benchmark in both single-target and nondegenerate multi-target scenarios. We further reveal a space-time sampling tradeoff under the same physical aperture. Based on the proposed architecture, we formulate a non-convex joint resource allocation problem that maximizes the weighted downlink-uplink sum rate by jointly designing the sensing transmit covariance, downlink precoder, and uplink receive beamformers under sensing accuracy, BS transmit-power, communication QoS, and residual SI constraints. An alternating-optimization-based algorithm is developed. Simulations demonstrate consistent performance gains over the considered baselines and confirm the complementary benefits of the CPA virtual aperture and sensing covariance optimization.

cs.IT↗

arXiv2026

Comments on the recent improvements of the MRRW bounds

The asymptotic McEliece--Rodemich--Rumsey--Welch bound (1977) limits the largest attainable rate of binary codes as a function of the relative distance. After a nearly half-century hiatus, this result was recently improved in two concurrent works, by OpenAI and by O. Alrabiah and V. Guruswami. The two arguments look entirely different, a Delsarte certificate on the one hand, a classical-quantum channel and the pretty good measurement on the other, and they yield the same bound. The purpose of this note is to explain why: in both proofs, a subspace is attached to every codeword and moved with it, and the bound counts how many such subspaces fit in the ambient space, exactly in the first case and in the probabilistic sense of typicality in the second. We also present the OpenAI proof in the language and context of coding theory, as an extension of the spectral method in which the single vector attached to a codeword is replaced by a subspace.

cs.IT↗

arXiv2026

Entropy lower bounds and sum-product phenomena

Various lower bounds are established for the entropy of sums, products and their combinations. First, we derive a prime-field analogue of a version of the entropy power inequality established by Tao over torsion-free groups. Next, we prove an entropy sum-product statement: For independent and identically distributed random variables $X,X'$, the maximum of ${\bf H}(X+X')$ and ${\bf H}(XX')$ is bounded below by a linear combination of the entropy and the min-entropy (Rényi entropy of order~$\infty$) of $X$. This result, obtained by bounding entropies of the form ${\bf H}\bigl( X(Y+Z)\bigr)$ from above and below, is valid over arbitrary fields $F$. Over $F={\bf R}$, a slightly stronger inequality is derived. Finally, a weak version of a purely Shannon-entropic sum-product result is developed: If the entropic additive doubling of a random variable $X$ over an arbitrary field is $O(1)$, then its multiplicative doubling is at least proportional to ${\bf H}(X)$.

math.CO↗

arXiv2026

Moments of crosscorrelation demerit factors of binary sequences

Families of sequences with low mutual aperiodic crosscorrelation assist the design of systems for multi-user asynchronous communications and multiple-input multiple-output radar. The crosscorrelation demerit factor of a pair of sequences is the sum of the squared magnitudes of their crosscorrelation values at every shift when the sequences are normalized to unit Euclidean norm, and the merit factor is the reciprocal of the demerit factor. For each positive integer $\ell$, we endow the $2^{2 \ell}$ pairs of binary sequences of length $\ell$ with uniform probability measure and study the distribution of their crosscorrelation demerit factors. Sarwate showed that the mean value is always $1$ regardless of length $\ell$. We develop a method for finding an exact formula for the $p$th central moment (for any positive integer $p$) as a function of $\ell$. Formulae for the variance and third central moment ($p=2$ and $3$) are then obtained by hand calculations, while the fourth through sixth central moments are obtained by computer-assisted calculations. Our theory also shows that all the central moments must be strictly positive for $p\geq 2$ and $\ell \geq 3$.

cs.IT↗

arXiv2026

On two proofs of $d^2$ mixing of weighted Dikin walks

We study the mixing time of weighted Dikin walks for sampling from exponential distributions on polytopes and truncated positive-semidefinite (PSD) cones. Our first result gives a general total-variation mixing bound under strong self-concordance, $\barν$-symmetry, and mixed-trace regularity on the local metric. The key idea is to control the Metropolis--Hastings acceptance probability on a high-probability region rather than at every point. Applying this framework to the Lee--Sidford, Lewis-weight, and John metrics yields an $\widetilde O(d^2)$ mixing bound for sampling from polytopes, while applying it to a hybrid barrier yields an $\widetilde O(d^4)$ mixing bound for sampling from truncated PSD cones. Our second result establishes stronger $χ^2$-divergence guarantees and pointwise acceptance control using a new fourth-order bootstrap condition. For a suitably scaled Lee--Sidford metric, this yields an $\widetilde O(d^2)$ mixing bound in $χ^2$-divergence, improving on the previous $\widetilde O(d^{9/4})$ bound.

cs.DS↗

arXiv2026

Exponential random graph models with soft clique constraints

Let $r\geq3$ be fixed, and let $\mathbf{G}_n$ be the set of all simple graphs with vertex set $[n]=\{1,\ldots,n\}$. We consider an exponential random graph model which gives higher probability to $G \in \mathbf{G}_n$ than to $H \in \mathbf{G}_n$ if $G$ has fewer $r$-cliques than $H$. But all graphs in $\mathbf{G}_n$ have positive probability. The degree to which graphs with fewer $r$-cliques are given higher probability is determined by a positive weight $w$. We prove that, asymptotically almost surely as $n \to \infty$, a random graph from $\mathbf{G}_n$ has a vertex partition into $r-1$ parts of roughly equal size, the density of edges between the parts is close to $1/2$, and for every $\varepsilon > 0$ the density of edges within any part is less than $\varepsilon$. The asymptotic structural properties are independent of the weight $w$ as long as it is positive. We also extend the result to the context of several clique sizes, each one with its own weight.

math.CO↗

arXiv2026

Fine Difference Structure and Prime-Power Depth of Bent Partitions

A $p$-ary bent partition of $\mathbb{F}_p^n$ is a partition into $K$ nonempty cells such that every balanced assignment of its cells to $\mathbb{F}_p$ produces a bent function. It was asked whether every possible depth $K$ is a power of $p$; for general $p$, previous affirmative results required regularity or cell-symmetry hypotheses. We prove the stronger unconditional statement that, for every nonzero $h$, exactly $p^n/K$ points remain in the same fine cell under translation by $h$. Thus the fine cells form a partitioned difference family and the fine label map is zero-difference balanced. Consequently $K\mid p^n$, so $K=p^t$; nonempty cells further give $1\le t<n$. In even dimension, the classical cell-size theorem yields $K\mid p^{n/2}$. Together with the known odd-dimensional ternary three-fibre parameter restriction, this gives the global bound $t\le\lfloor n/2\rfloor$. The proof is an exact finite average over balanced coarsenings. The main counting identity and selected consequences are formalized and kernel-checked in Lean 4.

cs.IT↗

arXiv2026

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↗

arXiv2026

Towards a mathematical theory of superposition

We develop a mathematical theory of superposition in neural networks using tools from frame theory and compressed sensing. In our model, a sparse binary vector $x$ of active features is encoded through an overcomplete dictionary $W$, and feature recovery is performed by applying $\operatorname{ReLU}(W^\top W x+b)$ with an appropriate bias vector $b$. We prove several recovery theorems for this model. In the random-support setting, we establish high-probability support recovery for nearly tight, low-coherence dictionaries, with guarantees when the expected sparsity is up to order $d/\log n$. In the worst-case support setting, we give a sharp and computable criterion for which sparsity levels permit support recovery. We apply this criterion to Gaussian random matrices and equiangular tight frames. For real equiangular tight frames with $n>d+1$, we determine the exact recovery threshold in terms of the coherence. The proof of this result for real equiangular tight frames relies on a novel characterization---which should be of independent interest to frame theorists---of the distribution of signs in the Gram matrix.

stat.ML↗

arXiv2026

Beyond the Bethe Approximation of the Permanent

The canonical Bethe approximation gives a deterministic approximation to the permanent of every nonnegative matrix within a factor of $(\sqrt{2})^n$. We improve the base of this exponential factor: for some absolute constant $c<\sqrt{2}$, there is a deterministic polynomial-time $c^n$-approximation for the permanent of every nonnegative matrix. This shows that the canonical Bethe guarantee is not a barrier for deterministic approximation of the permanent. The proof augments the Bethe lower bound with a new certificate tailored to matrices on which that lower bound loses nearly the full factor. The author supplied the high-level plan of attack, and the proof was developed in an interaction with ChatGPT 5.6 Sol Pro. The author subsequently verified the results. Codex assisted with proof checking, manuscript assembly, and typesetting.

cs.DS↗

arXiv2026

The Class Edge-Reconstruction Number of a Maximal Planar Graph Is One or Two

An edge card of a graph is obtained by deleting one edge, and a class edge-reconstruction number asks for the fewest carefully selected cards that identify the graph when its class is known. We determine the sharp universal bound for maximal planar graphs. Two selected cards always suffice, and the octahedral graph shows that two can be necessary; some maximal planar graphs are already identified by one card. The argument exploits the fact that deleting a flippable edge leaves a single quadrilateral whose two diagonals give the only possible maximal-planar completions. Degree information then rules out the competing completion, with a separate argument for graphs containing a vertex of degree three. This settles a problem posed in a 2010 survey on reconstruction numbers.

math.CO↗

arXiv2026

Algebraic Geometry Codes Approach the Half-Singleton Bound with Constant Field Size

We study linear codes for insertion and deletion (insdel) errors through the lens of evaluation codes. We develop a general framework for analyzing random puncturings of evaluation codes, where the edit distance is controlled by only the size of the evaluation domain and the maximum number of zeros of a nonzero function in the underlying function space. Our proof generalizes the results of Con, Guo, Li, and Zhang (ICALP 2025), and simultaneously simplifies their arguments by avoiding an in-depth analysis of longest common subsequences. We demonstrate the applicability of our core theorem by instantiating it with random puncturings of Reed--Muller codes. We then recover the result that random Reed--Solomon codes approach the half-Singleton bound over linear-sized fields while also improving the dependence on the additive gap $\varepsilon$ from $2^{O(1/\varepsilon^2)}$ to $2^{O(1/\varepsilon)}$. Finally, by applying the framework to algebraic geometry codes arising from asymptotically good towers of function fields, we show that there exist randomized families of structured linear codes over constant-sized fields that approach the half-Singleton bound.

cs.IT↗

arXiv2026

Algorithmic exploration of the unit distance problem in the rational plane

This paper presents reproducible experimental evidence on unit-distance graph density. Our approach is based on a novel algorithmic exploration of the rational plane for the generation of unit-distance graphs. An efficient algorithm for this utility must perform a local-breadth search on a bounded and finite set of elements and generate a graph that potentially encompasses the general properties of a unit-distance graph, not affected by restrictions on its generation. To this end, we show that our approach accomplishes this purpose by overcoming the limitations of grid-based structures, used in the literature for generating unit-distance graphs. When applied to large unit-distance graphs, our method generates large graphs with a scaling exponent of 1.17, that surpasses recent theoretical predictions on graph density growth.

cs.CG↗

arXiv2026

Group-averaged Markov chains II: tuning of group action in finite state space

We study group-averaged Markov chains obtained by augmenting a $π$-stationary kernel $P$ with orbit kernels induced by a group action. We analyse the Gibbs ($G$), Metropolis--Hastings ($M$), and Barker ($B$) kernels, their sandwiches $QPQ$, and mixtures $\tfrac{1}{2}(P+Q)$, where $Q\in\{G,M,B\}$. Under suitable conditions, $M^t$ and $B^t$ converge blockwise to $G$. The projection chains of $GPG$ and $P$ coincide, while every sandwich $QPQ$ has absolute spectral gap no smaller than that of reversible $P$. For $GPG$, we derive an additive asymptotic-variance bound, prove monotonicity for $G$-invariant observables, and identify it as the Kullback--Leibler (KL) information projection of $P$ onto the $G$-invariant kernels. For a fixed orbit partition, the spectral and KL properties of $GPG$ reduce to those of a lower-dimensional orbit-space chain. Among Gibbs projections with a prescribed number of orbits, we identify the partition minimizing KL divergence to stationarity and characterize exact stationarity. Finally, alternating group projections converge at a rate determined by singular values of an overlap matrix and, in structured cases, can yield exact sampling with logarithmically many group actions. These results motivate tuning heuristics and yield polynomial mixing for a Curie--Weiss example in a regime where Glauber dynamics is exponentially slow.

math.PR↗

arXiv2026

Divisibility of Trace Codes

A linear code is said to be $Δ$-divisible if the Hamming weights of all its codewords are divisible by $Δ$. The $p$-adic valuation of a code is defined as the greatest integer $t$ such that the code is $p^t$-divisible. In this paper, we establish a divisibility criterion for trace codes. Specifically, this criterion provides a systematic method to determine the $p$-adic valuation of the associated trace code, thereby extending Ward's classical divisibility criterion from standard generating sets (or matrices) to generalized generator matrices over an extension field. Furthermore, we present two applications of our framework. The first application provides a concise proof of the celebrated divisibility results on semisimple Abelian codes established by Delsarte and McEliece. The second application establishes several explicit lower bounds on the $p$-adic valuation of the number of solutions over $\mathbb{F}_{q^m}$ (where $q = p^e$) to the Artin--Schreier type equation $ f(x_1,\ldots,x_k)=y^q-y $. In particular, under the coprime condition $\left(d,\frac{q^m-1}{q-1}\right)=1$, we determine the exact minimum $p$-adic valuation of the number of solutions when $f$ is restricted to homogeneous polynomials of degree $d$.

math.CO↗

arXiv2026

Optimal girth-dependent bounds for the Bethe approximation of the permanent

For an $n\times n$ nonnegative matrix $A$, the Bethe permanent, which is computable in deterministic polynomial time, satisfies the tight universal comparison \[\operatorname{Bethe}(A) \leq \operatorname{per}(A) \leq 2^{n/2}\operatorname{Bethe}(A).\] The lower bound, due to Gurvits, is attained on forests. The upper bound, due to Anari and Rezaei, is attained by the adjacency matrix of a disjoint union of $4$-cycles. Confirming a conjecture of Anari, we provide an optimal girth-dependent refinement of the above comparison. More precisely, we show that if the bipartite support graph of $A$ has girth at least an even integer $g \geq 4$, then \[\operatorname{Bethe}(A) \leq \operatorname{per}(A) \leq 2^{2n/g}\operatorname{Bethe}(A).\] The upper bound is attained by the adjacency matrix of a disjoint union of $g$-cycles.

math.CO↗

arXiv2025

The differential properties of certain permutation polynomials over finite fields

Finding functions, particularly permutations, with good differential properties has received a lot of attention due to their varied applications. For instance, in combinatorial design theory, a correspondence of perfect $c$-nonlinear functions and difference sets in some quasigroups was recently shown by Anbar et al. (J. Comb. Des. 31(12):1-24, 2023). Additionally, in a recent manuscript by Pal et al. (Adv. Math. Communications, to appear), a very interesting connection between the $c$-differential uniformity and boomerang uniformity, when $c=-1$, was pointed out, showing that they are the same for an odd APN permutation, sparking yet more interest in the construction of functions with low $c$-differential uniformity. We investigate the $c$-differential uniformity of some classes of permutation polynomials. As a result, we add four more classes of permutation polynomials to the family of functions that only contains a few (non-trivial) perfect $c$-nonlinear functions over finite fields of even characteristic. Moreover, we include a class of permutation polynomials with low $c$-differential uniformity over the field of characteristic~$3$. To solve the involved equations over finite fields, we use various number theoretical techniques, in particular, we find explicitly many Walsh transform coefficients and Weil sums that may be of an independent interest.

math.CO↗

arXiv2026

Binary Multiple-Node-Erasure-Correcting Codes over Complete Graphs: Constructions, q-Ary Metric Balls, and Duality

We study linear codes whose coordinates are the ordinary edges and self-loops of complete undirected graphs; a node erasure removes all coordinates incident with a failed vertex. The construction results are binary. For triple-node erasures, we extend the published cyclic construction by allowing a suitable cyclic check slope to depend on the prime graph length. An explicit determinant test proves that one of three fixed slope choices works at infinitely many prime lengths, unconditionally, and gives redundancy $3n-2$, one bit above the graph Singleton bound. We also give Singleton-optimal triple-node codes at $n=6,8,10,12$, together with a general ordinary-edge framework that isolates the remaining loop-completion problem. When $2$ is primitive modulo an odd prime $n$, a binary multi-slope construction corrects every $ρ$-node erasure for $2\leqρ<n$, with redundancy $ρn-(ρ-1)$ in the range $2\leqρ\leq(n+1)/2$. Returning to arbitrary prime powers, we derive exact generating transforms and inclusion--exclusion formulas for node-metric ball volumes, fixed-radius asymptotics, and packing, existence, and covering bounds. Finally, for the complementary clique-erasure metric, we obtain an exact weight enumerator and a Singleton-optimal node--clique duality.

cs.IT↗