Papers

The classical Grünbaum inequality asserts that the proportion of the volume of a convex body cut off by a halfspace containing its barycenter is at least $1/e$. From its functional counterpart, for any log-concave random variable $X$, one has $\mathbb{P}(X\ge \mathbb{E}X)\ge 1/e$, with equality if and only if $X$ is exponential. Motivated by Grünbaum's inequality for convex bodies and its functional generalizations, we prove analogous inequalities for entropy, with characterizations of the equality cases. We show that if $X$ is a log-concave random variable on $\mathbb{R}$, then $$ h(X)-\frac{e}{e-1}H_2(1/e) \leq h(X|X \leq \mathbb{E}X) \leq h(X), $$ where $h$ is the differential entropy, $H_2(\cdot)$ is the binary entropy function and $X|X\leq \mathbb{E}X$ stands for the distribution of $X$ conditional on $X\leq \mathbb{E}X$. We generalize the upper bound for all Rényi entropies and the lower bound for min-entropy. Our inequalities are sharp and we characterize all equality cases. We discuss potential generalizations in high dimensions and give counterexamples in some directions. As an intermediate step for the proof of the lower bound, we establish a new inequality that we prove using a technique known as degrees of freedom, combined with a standard KKT-type optimization lemma. Along the way, we characterize the equality case in a known comparison inequality between differential and min-entropy, which may be of independent interest.

In this paper, we study the multi-target detection problem in a movable-antenna (MA)-enabled wireless sensing system with linear arrays, in which both the direct line-of-sight (LoS) paths and the first-order non-LoS (NLoS) paths are explicitly considered. Unlike conventional fixed-position antenna arrays, MAs provide additional design degrees of freedom by enabling adaptive antenna positioning to reconfigure the propagation geometry, offering great potential to enhance the sensing performance in complex multi-target multipath scenarios. Under this setup, we first develop a cross sparsity Markov mixture prior to derive the posterior probabilities of target locations, in which the structural correlation between the LoS and NLoS paths is effectively exploited to enhance the location estimation accuracy. Based on the derived posterior probabilities, we further analyze the angular-domain sensing performance for multi-target detection by proposing a new two-dimensional (2D) ambiguity function as the performance metric. Next, we optimize the MA positions to suppress the sidelobe levels and narrow the mainlobe width of the proposed ambiguity function. Although the resulting problem is highly non-convex, we develop a low-complexity Dykstra-based projected gradient descent algorithm to solve it efficiently. Finally, simulation results verify the accuracy of the proposed ambiguity function analysis, demonstrate the substantial performance gains enabled by the proposed prior model and MAs, and show that the proposed algorithm achieves performance comparable to existing methods with significantly lower computational complexity.

Low-resolution quantization constrains the maximum achievable gains of multiple-input multiple-output (MIMO) systems. While the adverse effects and mitigation strategies have been thoroughly analyzed for communication systems, the impact of low-resolution quantization on integrated sensing and communication (ISAC) systems remains insufficiently explored in the existing literature. In this paper, we propose an analysis and design framework to investigate and mitigate the effects of 1-bit digital to analog converters (DACs) for ISAC systems. Firstly, an analytical sensing signal-to-noise ratio (SNR) expression is derived by using the Bussgang decomposition. Furthermore, two different methodologies are proposed to design a transmit waveform that satisfies both communication and sensing requirements simultaneously. The first method uses a separate constant modulus (CM) sensing signal since CM signals are known to be more robust to nonlinear distortion than orthogonal frequency division multiplexing (OFDM) modulated signals. The second method employs the squared-infinity norm Douglas-Rachford splitting (SQUID) approach to construct the transmit waveform using nonlinear quantized precoding. Finally, the performance of the proposed methods are validated via numerical simulations to indicate the complexity-performance tradeoff between two different methods.

Driven by the exponential growth of latency-sensitive applications, mobile edge computing (MEC) has emerged as a pivotal paradigm, yet mitigating its substantial energy consumption remains critical. This paper explores a movable-antenna (MA) assisted energy minimization scheme in an uncrewed aerial vehicle (UAV)-enabled MEC system, where a UAV equipped with an MA array serves as an edge server to process tasks offloaded from terrestrial consumer electronics (CE) devices. To minimize the total system energy consumption, we jointly optimize computation resource allocation, CE transmit power, receive beamforming, and MA positions. To tackle the resulting non-convex problem with coupled variables, a robust alternating optimization algorithm based on the block coordinate descent method is developed. The problem is iteratively decomposed into three subproblems. In particular, the subproblem of transmit power and receive beamforming is reformulated and optimized using the quadratic transform technique, while the MA array positions are optimized via the particle swarm optimization algorithm. Numerical simulations verify that the proposed scheme achieves substantial energy savings over conventional benchmarks.

As 6G wireless networks transition toward sub-Terahertz (sub-THz) frequencies to satisfy extreme capacity demands, managing the trade-off between massive bandwidth and power consumption becomes a critical design challenge. In this paper, we investigate the fundamental energy efficiency (EE) limits of a dual-band base station site combining a coverage-oriented sub-6 GHz carrier with a capacity-oriented sub-THz carrier. By jointly optimizing hardware parameters and advanced sleep modes via activity factors, we identify four distinct operational regions that govern the EE-optimal behavior across the complete range of data rates. We derive closed-form analytical thresholds that dictate precisely when the sub-THz band should awaken from sleep and how to allocate traffic between the bands in that case. Our results demonstrate that utilizing the sub-THz band is EE-optimal when the power cost of the bandwidth-limited sub-6 GHz band surpasses the static power penalty of activating the sub-THz circuitry. Ultimately, this framework provides mathematically rigorous guidelines for power consumption minimization and sleep-mode management in future green networks.

Goppa codes play an important role in code-based cryptography due to their efficient decoding algorithms and their use as underlying private codes in the McEliece cryptosystem. To determine the true minimum distances of Goppa codes is a notoriously difficult problem. In this paper, we establish a criterion for a Goppa code to attain its designed distance. We consider Goppa polynomials of the form $G(x)=U(x)H(x)+V(x)H'(x)$, where $\deg(G)=t$ and $H(x)\in\mathbb{F}_q[x]$ is a monic irreducible polynomial of degree $t+1$ whose roots are contained in the support $L$. We prove that the corresponding Goppa code $\Gamma(L,G)$ contains a codeword of weight $t+1$ if and only if \[ \frac{V(\alpha_{i_{t+1}})}{V(\alpha_{i_j})}\in\mathbb{F}_q^*, \qquad 1\leq j\leq t, \] where $\alpha_{i_1},\ldots,\alpha_{i_{t+1}}$ are the roots of $H(x)$. Based on this criterion, we derive a general family of Goppa codes that attain their designed distance by developing an interpolation-based construction of Goppa polynomials. We further obtain families of Goppa codes whose Goppa polynomials are determined by considering monomial, binomial, and their product of the auxiliary polynomial $V(x)$. By taking $H(x)$ to be different irreducible binomials and trinomials, we obtain several explicit families of Goppa codes whose minimum distances are equal to designed distance.

Classical multi-user diversity theory predicts that throughput over Rayleigh fading channels grows as $\log_2\!\log_2 K$. In this work, we demonstrate a fundamental shift in this scaling law under heavy-tailed composite fading. Specifically, under Fisher–Snedecor $\mathcal{F}$ composite fading, the channel power acquires a regularly varying upper tail, shifting extreme-value statistics from the Gumbel to the Fréchet domain. We prove that the maximum SINR among $K$ users scales polynomially as $K^{1/m_s}$, where $m_s$ is the shadowing severity parameter, leading to an ergodic capacity scaling of $\frac{1}{m_s}\log_2 K$. Crucially, this scaling persists in interference-limited Poisson networks, where aggregate co-channel interference alters the scaling constant but not the exponent. This polynomial gain is most relevant in severe-to-moderate shadowing ($m_s \le 3$), as encountered in body-area networks, vehicular/industrial IoT, and dense indoor environments, where Fréchet asymptotics overtake industry-standard lognormal models at practical user counts. Finally, we establish the conditions necessary to harvest this gain (showing that proportional-fair scheduling under quasi-static shadowing reverts to Gumbel scaling) and validate all analytical findings through Monte Carlo simulations, including MIMO random beamforming.

Quantum devices are open systems whose dynamics interleave coherent evolution with dissipation, and benchmarking, error mitigation, and error correction all rest on a faithful model of both. Existing characterization protocols either assume prior knowledge of the interaction and noise structure, or demand ancillas, entangled probes, or mid-circuit control, or capture only the Pauli-diagonal part of the noise. Here, we present a protocol that reconstructs an arbitrary sparse Markovian generator, including every Hamiltonian together with the jump operator coefficients, using only product Pauli state preparation, single uninterrupted forward evolutions, and product Pauli measurements. Given a sparsity budget $M_0$ and a strength bound $\Gamma$ of the Lindbladian, every coefficient is learned to precision $\epsilon$ from $\widetilde{O}(\Gamma^2M_0^2/\epsilon^4)$ experiments and $\widetilde{O}(\Gamma M_0^2/\epsilon^2)$ total evolution time, with both supports identified from data without locality assumptions. The protocol runs at a logarithmic number of positive evolution times on a hardware clock lattice and is provably robust to calibrated state-preparation and measurement errors.

Accurate and real-time direction of arrival (DOA) estimation is crucial for beamforming in unmanned aerial vehicle (UAV) communication systems. However, the existing high-precision DOA estimation algorithms encounter high computational complexity when implemented on a UAV with on-board signal processing constraints. To tackle this issue, an electromagnetic neural network (EMNN) is developed for DOA estimation, which is capable of generating the angular spectrum of the incident signal based solely on amplitude observation. Specifically, the proposed EMNN consists of two components: a stacked intelligent metasurfaces (SIM) is mounted on the UAV, and each meta-atom is an artificial neuron that can process signals in the electromagnetic domain with low energy consumption and ultra-fast computing speed. Furthermore, a fully connected layer is cascaded to process the received amplitude signal, enhancing the non-linear extraction and representational ability of EMNN. Moreover, to reduce the computational complexity and observation snapshots required for high-resolution DOA estimation, we develop a hierarchical DOA estimation framework, which involves two stages for conducting coarse and fine DOA estimation, respectively. For each stage, EMNN is trained on randomly generated training samples and their corresponding spectra to achieve the desired estimation goal. Finally, the simulation results validate that the proposed EMNN achieves approximately 13 dB gain in classification error reduction over the conventional beamforming (CBF) method in dual-signal scenarios, albeit its lower cost and radio frequency (RF)-related power consumption.

We develop a theory of approximate quantum error correction (QEC) based on the error-set model, complemented by general methods for code construction. Exact QEC has a powerful error-set structure: by the Knill-Laflamme conditions, a code correcting a given error set automatically protects against every channel whose Kraus operators lie in their linear span. This linearity gives rise to code distance, the equivalence between erasures and general errors, and a theory of asymptotically good codes. A longstanding view has been that these features do not extend to AQEC, leaving the theory essentially channel-by-channel. We show instead that, although full Knill–Laflamme linearity fails, a restricted form survives and suffices to extend all three structural features to the approximate setting. Specifically, a common error-set criterion governs families of channels whose Kraus operators are linear combinations of a given error set and whose coefficient matrices satisfy a spectral constraint. Using the Bény-Oreshkov worst-case and Petz average-case frameworks, we derive uniform fidelity guarantees for these families in terms of two new code parameters–the environment-leakage distance, controlling worst-case performance, and the Knill-Laflamme Hellinger distance, characterizing the average-case performance of Petz recovery. To demonstrate the scope of this model, we develop partition-based constructions across diverse quantum systems and geometries, placing exact and approximate correction on equal footing. These constructions lead to a metric–error alignment hierarchy for Hilbert spaces, metrics, and error families, which in turn characterizes the resulting recovery guarantees. They yield the first known asymptotically good code families for fermionic systems, one-dimensional Rydberg-blockaded systems, and deletion errors, and extend to other physical platforms.

External difference families (EDFs) are closely related to algebraic manipulation detection (AMD) codes, a cryptographic primitive that protects messages against additive tampering by an adversary who cannot observe the transmitted codeword. We first study a cyclotomic construction of external partial difference families over finite fields and derive a necessary and sufficient condition under which the resulting families are EDFs. For even block sizes up to $14$, we obtain explicit criteria in terms of quadratic and biquadratic residues, yielding new $R$-optimal weak AMD codes. We then construct bounded generalized strong external difference families using cyclotomic classes over finite fields, direct products of finite fields, and generalized cyclotomic classes over integer residue rings. These constructions give infinite families of systematic $G$-optimal strong AMD codes with flexible parameters.

There are eleven finite rings of order $p^{2}$ denoted by $A_p$ to $K_p$ in alphabetical order. In particular, we consider $I_{2}$ and $I_{3}$ which are commutative non-unital rings of orders 4 and 9 defined by generators and relations as \[I_{p}=\left\langle a,b\mid pa=pb=0,\:a^{2}=b,\:ab=0\right\rangle\] for $p=2, 3,$ respectively. Alahmadi et al. studied codes over these rings. In this paper, we study additive complementary dual (ACD) codes over the rings $I_{2}$ and $I_{3}$. We show relations between ACD codes over $I_{2}$ and binary linear complementary dual (LCD) codes using a reduction map from $I_{2}$ to $\mathbb{F}_{2}$, and between ACD codes over $I_{3}$ and ternary LCD codes using a reduction map from $I_{3}$ to $\mathbb{F}_{3}$. Using the first relation, we classify ACD codes over $I_{2}$ with the highest minimum distances for $n=1, 2, 3$ and partially for $n=4, 5$. It turns out that they are maximum distance separable (MDS) codes. Using the second relation, we classify ACD codes over $I_{3}$ with the highest minimum Lee distances for $n=1, 2$ and partially for $n=3$. We generalize the two relations into a relation between ACD codes over $I_{p}$ and $p$-ary LCD codes using a reduction map from $I_{p}$ to $\mathbb{F}_{p}$. This is a correction of the paper published in Advances in Mathematics of Communications, Volume 24, pages 61-76, 2026. In particular, we corrected the statements of Theorems 3.5, 4.7, 4.8, and their proofs.

We review the renormalization group framework for signal detection in high-dimensional data, tailored to the regime where the signal may be of extensive rank and does not separate from the noise bulk as isolated spikes. The framework provides a conceptually simple criterion for distinguishing signal from noise within a quasi-continuous spectral region near a random-matrix universality class. This scenario lies beyond the reach of standard methods such as the Baik-Ben Arous-Péché threshold, which requires eigenvalues to be cleanly separated from the bulk. The renormalization group approach, by contrast, directly tracks spectral deformations and consistently yields a lower limit of detection without relying on spike separation. We review results that identify the presence of a signal by testing the stability of the Gaussian fixed point of an effective field theory for the collective behaviour of the degrees of freedom in the spectral tail, where the signal resides. We also discuss how the scale dependence of the canonical dimension, induced by the signal, manifests as a dimensional phase transition.

We develop a framework for quickest detection of distributional change in signals modeled as rough paths. By representing the pre-change and post-change dynamics as two independent rough paths and modeling the observed signal as their concatenation at an unknown random change-point, we formulate quickest detection as an optimal stopping problem using rough paths. We show that the optimal stopping rule takes the form of the first hitting time to a half-space by a linear functional of the rough path signature, and establish that the same structural form arises independently from the intrinsic geometry of the observed path. Statistical guarantees on detection delay and false alarm probability are derived, and the framework is extended to a distributionally robust formulation in which both the pre-change and post-change models are uncertain. The proposed rules are implemented via zeroth-order stochastic approximation over truncated-signature coefficients and evaluated numerically under Brownian and fractional Brownian dynamics, achieving performance comparable to optimal methods in the Brownian setting while outperforming them and remaining robust to adversarial path perturbations in the fractional Brownian setting.

This manuscript addresses a connection between operations in coding theory and graph theory, as well as some operations in both areas. The concept of the Hasse diagram of a binary linear code, $\Gamma(C)$, is recall; this concept visually represents the cosets of a binary linear code with a partial ordering. The main objetive is to give a characterization of the graphs corresponding to: extended codes, punctured codes, and the direct sum of two codes, using the graph of the given binary linear code as a starting point. Relationships are established between the graphs of code operations and particular operations of graphs.

Future wireless systems will require ever larger antenna arrays and heavier signal processing, making conventional digital multiple-input multiple-output (MIMO) architectures difficult to scale. In this paper, we show that a possible solution is to offload part of the processing from the digital to the analog domain. This can be done through linear microwave networks designed to compute directly using the communication signals at radio frequency (RF). These networks, denoted as microwave linear analog computers (MiLACs), can perform useful matrix operations instantly through wave propagation. Remarkably, although MiLACs are linear, the output signals can depend nonlinearly on the tunable parameters of the network, enabling the computation of operations beyond simple linear transforms. In particular, MiLACs can realize matrix inversion and pseudo-inversion with complexity scaling quadratically with matrix size, rather than cubically, which is essential in zero-forcing beamforming. We then review how MiLAC-aided MIMO architectures can reduce the number of RF chains, relax the resolution requirements on digital-to-analog converters (DACs) and analog-to-digital converters (ADCs), and decrease the beamforming complexity. We finally discuss the main challenges related to MiLAC and promising directions for future research.

Coded computing is a reliable and fault-tolerant paradigm for executing large-scale computational tasks over distributed worker nodes. Among existing coded computing frameworks, Berrut Approximated Coded Computing (BACC) enables distributed computation of arbitrary non-polynomial functions through rational interpolation. Although BACC provides provable approximation guarantees and resilience against straggling workers, its robustness against Byzantine workers remains largely unexplored. To fill this research gap, we propose Robust Berrut Approximated Coded Computing (RBACC), which establishes a coding-theoretic framework for BACC by enabling error localization and error correction in the presence of Byzantine workers. In particular, RBACC introduces a new choice of evaluation points that establishes a connection between Berrut interpolation and Discrete Cosine Transform (DCT) codes, thereby enabling error localization and error correction under finite-precision arithmetic. We derive analytical upper bounds on the approximation error of RBACC under multiple operating scenarios, including straggler-only systems and systems with Byzantine workers under finite-precision arithmetic. Building upon this analysis, we formulate several optimization problems for selecting the DCT code dimension and for assigning encoded evaluations to unreliable workers. We show that these are previously unexplored design parameters that can be systematically optimized to improve the reconstruction accuracy. Experimental results demonstrate that the proposed RBACC framework effectively mitigates stragglers and Byzantine workers while offering improved reconstruction accuracy over the baselines.

The principle of maximum entropy provides a fundamental framework for characterizing typical structures of large random networks subject to observable constraints. In their pioneering numerical experiments \cite{radin2014asymptotics}, Radin, Ren, and Sadun conjectured that entropy-maximizing graphons satisfying subgraph density constraints are stochastic block models a conjecture we term the RRS conjecture. While several special cases have been proven for single-relation graphs with specific constraint families, the general problem has remained open, particularly for multi-relational networks. We resolve the RRS conjecture for constrained multi-relational graphons in the non-extremal regime, proving that entropy-maximizing solutions are step functions with finitely many blocks under the condition the subgraph density constraints are analytically independent and for almost all feasible combinations of sufficient statistics. Our proof employs a differential geometric technique to study solutions of constrained optimization problems in function space via functions with a finite parametrization (step functions). The two cornerstones of this work are: the generalization of subgraph density notion to $h$-subgraph density and the proof that manifolds that define the constrained region for the solutions maintain topological stability without developing new connected components under refinement. Together, these enable proving that no new global optima emerge in higher-dimensional spaces.

An additive $(n,k,d)_{q^m/q}$-code is a $\mathrm{GF}(q)$-linear subspace of $\mathrm{GF}(q^m)^n$ of $\mathrm{GF}(q)$-dimension $km$ with minimum Hamming distance $d$. We first extend the Alderson–Bruen–Silverman (ABS) model of linear codes to the additive setting: a code of length $n$ with $q^{km}$ words over an alphabet of size $q^m$ admits an ABS model if and only if it is equivalent to a nondegenerate additive code. We then ask whether an additive code that admits an extension must admit an additive extension. For linear codes ($m=1$) this is a theorem of Alderson and Gács. We characterize the additive codes admitting no additive extension as those whose associated projective system of flats is complete, and we prove that the answer to the question above is again affirmative for $(n,2,d)_{9/3}$-, $(n,2,d)_{4/2}$-, and $(n,3,d)_{4/2}$-codes. In contrast with the linear case, we show that the answer is negative in general. Scattered linear sets yield, for each square $q$, extendable additive $(n,2,d)_{q^2/q}$-codes admitting no additive extension. Further, a different method yields an extendable additive $(30,2,24)_{8/2}$-code with no additive extension. Consequently, for properly additive codes, completeness of the associated projective system does not imply maximality of the code. We conjecture that extendable $(n,2,d)_{p^2/p}$-codes, $p$ prime, always admit additive extensions.

Interference constitutes the fundamental performance bottleneck in wireless networks. Meanwhile, reconfigurable intelligent surface (RIS) has emerged as a promising technique for interference mitigation by directly modifying wireless channels. In this paper, we are interested in the following problem: whether interference-free transmission (in terms of Degree-of-Freedom, DoF) can be achieved with the aid of passive RIS in the two-way K-user interference channel, which is regarded as the most severely interfered network. We show that the answer is affirmative, i.e., interference in this network can be neutralized over the air. To accomplish this goal, two prominent challenges arise: i) the unit-modulus constraint on each RIS reflecting coefficient; ii) the significant disparity between the strengths of the direct and reflective channels. To address these challenges, we exploit the high-dimensional and random nature of wireless channels. Specifically, we cast the problem within a high-dimensional convex geometric framework, which enables us to leverage the ubiquitous concentration phenomenon in high-dimensional spaces. Based on this framework, we establish both sufficient and necessary conditions on the required number of RIS elements to achieve interference-free DoF, which turns out to coincide in order sense. Furthermore, we characterize the impact of imperfect channel state information (CSI) on the achievable DoF and show that interference-free DoF remains achievable if the CSI error is below a certain threshold. Simulation results validate our theoretical findings.

Interference fundamentally limits the performance of dense wireless networks, and reconfigurable intelligent surfaces (RIS) have recently emerged as a promising means of enabling interference-free transmission in the Degrees-of-Freedom (DoF) sense. This paper investigates the feasibility of achieving full DoF in a two-way K-user interference channel-a canonical interference-limited setting-by employing an active RIS. Unlike its passive counterpart, an active RIS is subject to both per-element gain constraints and a total reflection-power constraint, which renders over-the-air interference nulling equivalent to solving a constrained random linear system with coupled nonlinear constraints. By leveraging tools from high-dimensional convex geometry, we derive a tight scaling threshold on the required number of reflecting elements (REs) for full-DoF transmission. We further extend the analysis to scenarios where each RE incurs circuit power consumption under a total power budget, leading to a fundamental tradeoff between RIS transmit power and circuit power. For this setting, we establish the thresholds for both the total power and the corresponding number of REs required to achieve interference-free transmission. Simulation results validate the theoretical analysis.

Discrete-time Poisson channels with finite intersymbol interference provide a natural model for direct-detection optical links in which multipath memory and signal-dependent shot noise appear simultaneously. Under peak and average optical-intensity constraints, we study the independent and identically distributed (i.i.d.)-constrained capacity problem of such channels. We prove that every input distribution maximizing the stationary mutual information rate within the i.i.d. input class has finite support. The proof is carried out directly on the entropy rate of the continuous-state hidden Markov output process induced by the finite-memory channel. We first establish a filtering-forgetting estimate whose constants are uniform over all admissible i.i.d. input laws. We then derive the entropy-rate first variation, construct a holomorphic extension of the corresponding influence function, and combine the Karush-Kuhn-Tucker condition with a supralinear growth argument.

Score-based diffusion models have achieved remarkable empirical success in generative modeling, yet their approximation-theoretic foundations remain incomplete. In particular, although classical universal approximation theorems guarantee that neural networks can approximate score functions, it remains unclear whether such approximation guarantees translate into approximation of the probability distributions generated by reverse diffusion processes. In this paper, we establish a rigorous quantitative connection between these two notions. Specifically, we prove that if a neural network approximates the true score function sufficiently accurately, then the probability distribution generated by the corresponding reverse diffusion model is close to the target data distribution in Kullback-Leibler (KL) divergence, up to an irreducible mismatch between the terminal distribution of the forward diffusion process and the prior used to initialize the reverse process. More precisely, we derive an explicit upper bound on the distribution approximation error in terms of the score approximation error, the diffusion noise schedule, and the terminal prior mismatch. Our analysis combines Hornik's universal approximation theorem, Girsanov's theorem on path space, and the data processing inequality for relative entropy. Complementary to recent work that studies score approximation under finite-sample statistical settings and structural assumptions on the data distribution, our work develops an approximation-theoretic analysis based on classical neural network approximation theory. The resulting theorem provides a simple and explicit guarantee linking neural network approximation of score functions to approximation of the probability distributions generated by reverse diffusion models.

Binary cyclic codes are an important class of linear codes that admit highly efficient encoding and decoding algorithms. Constructing binary cyclic codes with favorable parameters has been an active research topic for several decades. In recent years, substantial progress has been made in the study of binary $[n,k,d]$ cyclic codes with $k$ close to $n/2$ and $d\geq \sqrt{n}$. Nevertheless, among the known codes of this type, only a few families simultaneously possess both large minimum distances and large dual distances. In this paper, for even $s$, we focus on new constructions of $[2^{s}-1,k]$ binary cyclic codes with rate approximately $1/2$, for which both the minimum distance $d$ and the dual distance $d^\perp$ are simultaneously large. Some of the constructions yield binary cyclic codes with parameters $[2^s-1,2^{s-1}\pm 1,d]$ such that the lower bounds of $d\cdot d^\perp$ are close to $n$ or $2n$. The other constructions yield binary cyclic codes with parameters $[2^s -1, 2^{s-1} + c, d]$ for some $ 1- \frac{5s}{2} \leq c \leq \frac{5s}{2} -1$ such that the lower bounds of $d\cdot d^\perp$ are close to $n$ or $2n$. Compared with the known binary cyclic codes of the same length and dimension, the majority of the binary cyclic codes we construct exhibit larger minimum distances. In particular, we obtain several codes with excellent parameters, which are verified against the Code Tables available at http://www.codetables.de/.

In multi-objective and multi-criteria decision-making under risk, especially in settings involving individual behavior, risk-aware analysis based on subjective evaluation has become increasingly important. Moving beyond risk-neutral modeling and the constraints of Expected Utility Theory (EUT), Cumulative Prospect Theory (CPT) provides a behaviorally grounded framework for capturing how individuals perceive and evaluate risky prospects. This paper conducts a rigorous theoretical analysis of Neilson's definitions of aversion. We provide a gamble-based interpretation, sharpen key conceptual distinctions, and make explicit the conditions under which the weak and strong notions coincide as well as when they diverge. Furthermore, we examine the Kobberling-Wakker utility function and related standard CPT specifications, highlighting structural limitations and inconsistencies that arise when these forms are required to satisfy Neilson-type aversion conditions. To address these issues, we propose a generalized CPT-utility function that retains the canonical reference-dependent shape while offering additional flexibility. This generalization extends the descriptive scope of CPT and provides an explicit functional form that is useful for sensitivity analysis and utility function-based optimization.

In analogy to the point-to-set principle for Hausdorff and packing dimension of Lutz, Lutz, and Mayordomo, we develop a Kolmogorov complexity point-to-set principle framework for upper and lower topological pressure and BS dimension in dynamical systems (and by extension for upper and lower entropy). As a demonstration of this framework we expand upon the Bowen pressure equation results of Climenhaga, as well as prove a pointwise version of the Bowen pressure equation.

The integrated sensing, communication, and computing (ISCC) system overcomes the limitations of conventional standalone architectures. Through resource sharing and collaborative design, it dynamically optimizes and jointly enhances communication, sensing, and computing performance, thereby significantly improving overall system efficiency. This work investigates the performance trade-off among secure communication rate, radar estimation rate, and computational energy efficiency in an uncrewed aerial vehicle (UAV)-assisted ISCC system. By jointly optimizing the UAV's three-dimensional (3D) trajectory, beamforming, user scheduling, and computational frequency, three optimization problems are formulated to maximize the average secrecy rate, sensing rate, and computational energy efficiency, respectively, thus establishing the system's performance boundaries under diverse scenarios. On this basis, the trade-off among security, sensing, and computation is further explored with the goal of maximizing the normalized weighted sum of the three performance metrics, which provides a theoretical basis for the performance-coordinated design of aerial ISCC systems.

Low Earth orbit (LEO) satellite constellations are poised to become a cornerstone of the sixth-generation (6G) Internet of Things (IoT), providing truly global coverage and ubiquitous connectivity. This article presents a holistic two-dimensional system architecture for 6G LEO satellite IoT that incorporates composition and functional perspectives to facilitate the seamless integration of LEO satellites and terrestrial networks. Building upon this architecture, we evaluate three pivotal enabling technologies targeting the uplink, downlink, and inter-satellite links (ISLs). Specifically, we analyze massive grant-free random access for efficient uplink connectivity, investigate deep learning-based multibeam precoding for robust downlink transmission, and examine distributed cooperative routing for resilient ISL data delivery. Furthermore, we present an on-orbit verification platform that validates the real-world feasibility and performance of the proposed solutions. Finally, we outline key open challenges and future research directions to guide the realization of future LEO satellite IoT.

The seamless integration of non-terrestrial and terrestrial infrastructures is a key enabler for ubiquitous connectivity in next-generation (NG) wireless networks. We investigate a hybrid satellite-cell-free Massive MIMO system, where multiple low-Earth-orbit (LEO) satellites jointly serve users in unison with terrestrial access points (APs) under realistic imperfect channel state information and practical user association constraints. We first derive closed-form expressions of the uplink ergodic throughput by exploiting maximum ratio combining (MRC) for transmission over spatially correlated Rician fading channels. Our analysis reveals the characteristic impact of both user-satellite and user-AP association patterns on both the spectral efficiency and rate-fairness achieved. We then formulate an energy efficiency optimization problem under joint user association and power control. Since the problems are inherently NP-hard due to the binary nature of the user-association variables, we develop an improved Differential Evolution (IDE) framework that efficiently explores the feasible solutions in polynomial time. Numerical results validate our analysis and show that the proposed hybrid scheme substantially improves energy efficiency and network throughput. For large-scale scenarios, the DE framework provides practical user-satellite-AP association guidelines, enabling scalable performance gains.

The Shannon capacity $\Theta(G)$ of a graph $G$ quantifies the maximum rate at which information can be transmitted with zero error over a noisy channel. It is lower bounded by $\alpha(G^d)^{1/d}$ for any $d$, where $\alpha(G^d)$ is the independence number of the $d$-th strong product of $G$. We construct independent sets of size $134753$ in $C_7^{10}$, $21909$ in $C_{11}^{6}$, $62530$ in $C_{13}^{6}$, and $8076974$ in $C_{15}^{8}$, improving the best known lower bounds for the Shannon capacity of these graphs to $\Theta(C_7)\geq 134753^{1/10}>3.258020$, $\Theta(C_{11})\geq 21909^{1/6}>5.289773$, $\Theta(C_{13})\geq 62530^{1/6}>6.300109$, and $\Theta(C_{15})\geq 8076974^{1/8}>7.301399$. We also improve the best known lower bounds on the independence numbers of several individual strong products of odd cycles that do not improve the Shannon capacity lower bound. The constructions were discovered through iterative interactions with a Large Language Model (LLM), illustrating the potential of LLMs for finding explicit combinatorial constructions.