Papers

We study the minimum-norm Gauss–Newton method for phase retrieval from random coded diffraction patterns with additive intensity noise. For bounded symmetric masks satisfying the standard complex moment conditions, we establish convergence guarantees with quadratic error recursions in both the resampled and fixed-data settings. In the resampled setting, $O(\log^2 n)$ masks per independent block suffice to obtain a spectral initializer within a constant-relative-radius neighborhood and to establish a local quadratic error recursion with an additive noise term. Consequently, in the noiseless case, the method achieves relative accuracy $\epsilon$ using $O(\log\log(1/\epsilon)\log^2 n)$ masks. In the fixed-data setting, $O(\log^6(en))$ masks suffice for spectral initialization within a neighborhood whose relative radius is proportional to $\log^{-5/2}(nL)$. Within this neighborhood, the method satisfies a linear contraction bound and eventually a quadratic error recursion. In particular, in the noiseless case, the method initially converges linearly and enters a quadratic convergence phase after $O(\log n)$ iterations, without acquiring additional masks. Numerical experiments on synthetic signals and a natural image demonstrate the method's efficiency relative to state-of-the-art algorithms.

The semantic entropy of a language model is the entropy of the meanings of its sampled answers. The meanings are never observed, and the entropy is estimated from the verdicts of a pairwise equivalence judge. Given the meanings of $n$ answers, a known channel declares each pair equivalent independently, with probability $q+\Delta$ within a meaning and $q$ across meanings. At most $K$ meanings occur, and $q$ is bounded away from $0$ and $1$. For a weak judge, with $n\Delta^2\to0$, the global minimax rate is set by distributions near a vertex of the simplex and overstates the difficulty elsewhere. We show that the difficulty is set by how many meanings can be rare. Over distributions with at most $s$ meanings below a fixed probability $\tau<1/(K-s)$, the minimax squared risk is of order $N^{-1/s}$ for $s\ge2$, $N^{-1}\log^2(e+N)$ for $s=1$ and $N^{-1}$ for $s=0$, capped at one, where $N=n^K\Delta^{2K-2}$. The matching lower bounds hold on every fixed polynomial window $n\Delta^2\le n^{-\kappa}$. One estimator attains these rates without knowing $s$ or $\tau$: it averages products of centered verdicts along all spanning trees of every $k$-set of answers, projects the resulting power sums onto the valid moment set and inverts them. The case $s=K-1$ gives the global rate $N^{-1/(K-1)}$ and the consistency threshold: with $\Delta=n^{-\gamma}$ the entropy is uniformly consistently estimable exactly when $\gamma<K/(2K-2)$. On polynomial windows, the same estimator attains the sharp local minimax constant on shrinking neighborhoods of interior points with distinct probabilities. Restricting the average to stars multiplies the constant by $K^{K-3}$.

Meeting hard task-completion deadlines for many simultaneously offloading users is the binding challenge in mobile-edge computing (MEC). As the user count grows, the transmit power required to satisfy every deadline within per-user budgets rises until no feasible allocation exists. We take a feasibility-first view of this $K$-user problem in an uplink non-orthogonal multiple access (NOMA) MEC system served by a pinching-antenna system (PASS), whose radiating points establish strong line-of-sight (LoS) links that lower the received-power requirement. We formulate a latency-constrained joint optimization of transmit powers, CPU frequencies, and antenna configuration. We then derive a closed-form least-power feasibility test: the componentwise-least allocation meeting every deadline. It certifies feasibility with the largest attainable margin to the power budgets and defines a practical quality-of-service (QoS)-feasibility outage metric, the probability that no allocation meets all deadlines within budget. We solve the problem with a three-block alternating-optimization algorithm that is provably monotone-convergent and has polynomial complexity. Numerical results show that the proposed scheme sustains deadline feasibility at user loads where equal-power, orthogonal multiple access (OMA), and single-antenna baselines fail or degrade sharply, each by a mechanism predicted directly from the system model, whereas the proposed design degrades gracefully. Stress tests further show that feasibility is sustained under order-of-magnitude budget reductions, and that the pinching-antenna array itself enables constrained-budget operation. The result is a scalable, closed-form-initialized design for deadline-feasible offloading.

Maximum-likelihood decoding of linear codes over erasure channels can be considered as solving a linear system of a column submatrix of the parity-check matrix. In practice, Gaussian elimination is usually applied, and its computational cost depends on the structure of the matrix, especially the sparsity and the fill-in generated during the elimination. The chordal graph, or triangulated graph, first introduced by C. Berge, has a so-called perfect elimination ordering, indicating that if the primal graph of a matrix is a chordal graph, then there exists a perfect elimination ordering such that a special class of fill-in under Gaussian elimination following this ordering will be avoided. In this paper, we introduce chordal codes, i.e., codes equipped with parity-check matrices such that the primal graphs of the parity-check matrices are chordal. We prove that, for a chordal code, the decoding submatrix associated with any erasure pattern retains a chordal primal graph. So, the complexity of chordal elimination (Gaussian elimination following the perfect elimination ordering applied to the chordal code decoding) will be smaller than decoding a random code by Gaussian elimination. Also, we provide an upper bound on the complexity of chordal elimination.

We prove that network coding and fractional routing have the same capacity region in undirected networks whose demand graph has matching number at most two, that is, when no three demand edges have pairwise disjoint endpoints. The metric-packing reduction of Liu et al. reduces this problem to a distance-weighted communication inequality on the fixed graph $\Gamma_{3,3}$. We first prove two entropy inequalities for sessions between designated terminal pairs on $\Gamma_{3,3}$. We then extend this result to sessions with arbitrary endpoints to obtain the distance-weighted communication bound required by the reduction. The proof combines conditional entropy inequalities with two graph contractions and an information inequality for deterministic interaction. Combining this inequality with the existing reduction establishes routing optimality for this class of demand graphs.

Does there exist an 8-variable balanced Boolean function with nonlinearity 118? This is a famous and long-standing open problem in Boolean functions. In this paper, we give a negative answer to this open problem, and prove that the maximum nonlinearity of balanced Boolean functions in 8 variables is 116, establishing the 8-variable instance of Dobbertin's conjectured recursion for the minimum Walsh peak of balanced functions. Equivalently, the covering radius of the Reed–Muller code RM(1,8) in 0-resilient Boolean functions is 116. We deduce the result by obtaining a quadratic form of polar rank 6, and reducing the problem to the existence of a 6-variable Boolean function with nonlinearity greater than 26 whose Hamming distance to a bent function is between 4 and 6.

What is the covering radius of the Reed–Muller code RM(1,9) ? This is a long-standing open problem in cryptography and coding theory. In this paper, we resolve this problem, and prove that the covering radius of RM(1,9) is 242. We exclude nonlinearity 244 by obtaining a cubic Boolean function achieving a sharp Walsh $\ell^1$ bound. Its equality cases reduce either to an impossible cubic hyperplane restriction or to two homogeneous cubic forms in the Brier–Langevin classification. To exclude nonlinearity 243, we deduce the inequality ${\Vert W_g\Vert }_{1}+2|W_g(0)|\le 10368$, and then reduce either to a spectrum bounded by 24 or to the same expression to be at least 10496.

Hybrid beamforming (HBF) reduces the hardware cost of massive MIMO base stations, but its channel acquisition requires recovering the full spatial-domain channel from highly compressed beam-domain observations. This paper proposes a learned score-based turbo message passing (L-STMP) framework for uplink HBF channel recovery under extreme spatial compression. By deeply unfolding the turbo message-passing loop and introducing trainable calibration parameters and learnable score priors in each layer, L-STMP liberates STMP from the idealized compression-matrix requirements of right-rotational invariance, and reduces inference complexity to only several unfolded layers. In addition, to preserve dominant channel components that random discrete Fourier transform (DFT) selection may miss, we introduce precoding matrix indicator (PMI)-assisted beam selection as a low-overhead directional prior. Simulation results demonstrate that L-STMP consistently outperforms existing baselines and obtains substantial reconstruction gains under PMI-assisted compression in channel reconstruction.

We investigate whether Rényi divergence can sharpen McDiarmid's inequality and its distribution-dependent refinement induced by KL divergence. Starting from Marton's transportation proof, we show that its transport inequality fails in general when KL divergence is replaced by Rényi divergence of order $0<\alpha<1$. For sums of independent bounded random variables, however, the additivity of Rényi divergence allows us to adapt the argument and derive concentration bounds. For Bernoulli sums, these bounds strictly improve on the distribution-dependent bound induced by KL divergence over a range of deviation thresholds. We also establish a distribution-dependent refinement of Gilardoni's inequality relating total variation to Rényi divergence, providing explicit constants for our general concentration bounds.

Watermarking embeds an imperceptible signal in text from large language models that is statistically detectable with a secret key. The recent textGrain watermark (Li et. al. (2026b)) calibrates this signal against the model's sampling entropy through an explicit entropy budget, combining the stochasticity of Red-Green and the unbiasedness of Gumbel-max watermarks, and its accompanying report proposes a detector that sums pivotal statistics. We study when, and how efficiently, textGrain can be detected after human edits, modeled as a sparse mixture in which a fraction $\varepsilon_n\asymp n^{-p}$ of positions retain the watermark and token uncertainty decays as $n^{-q}$. First, entropy calibration comes at a cost: the minimax detection boundary of textGrain is $p+q=1/2$, so its detectable region is strictly smaller than that of Gumbel-max and maximal-coupling watermarks, whose boundary is $2p+q=1$. This cost stems from textGrain's construction rather than from entropy calibration itself: an unbiased scheme with the same entropy loss remains detectable where textGrain is not. Second, perhaps surprisingly, this loss makes the sum detector optimal at the boundary when the number of transport columns is fixed, unlike for Gumbel-max. When the number of columns grows and the entropy budget does not vanish, the detectable region expands while that of the sum detector does not. Third, for a fixed edit fraction, we characterize the optimal exponential rate of type II error decay and show that the sum detector falls strictly short of it at every fixed alternative. Finally, we propose a weighted multiscale binomial scan, valid in finite samples and needing no knowledge of the next-token distributions, entropy budget, or edit fraction; it attains every detection boundary above and the optimal exponent.

Ferrers diagram rank-metric codes are essential components of the multilevel construction of constant dimension codes for random network coding. This paper aims to construct optimal Ferrers diagram rank-metric codes, focusing on the diagram \([3,3,3,5]\) with rank distance \(3\). We construct an optimal \([[3,3,3,5],6,3]_q\) Ferrers diagram rank-metric code for every prime power \(q\), thereby answering an open problem posed by Etzion et al. in 2016~\cite{Etzion2016}. The construction first builds a \([3\times4,4,3]_q\) MRD code and then proves that the Delsarte dual of its puncturing on the last column contains a \([3\times3,3,3]_q\) MRD code, which comes from semifield codes. We further construct optimal Ferrers diagram rank-metric codes for two infinite families of Ferrers diagrams: the first relies on the \([[3,3,3,5],6,3]_q\) construction and holds for every prime power \(q\), while the second is binary and obtained directly by the puncturing-inclusion construction.

We present a perceptual audio coder in which the entropy-coding model and the bit allocation co-adapt across blocks without model side information. Mantissas are coded by a hybrid of bit packing and arithmetic coding with per-band probability models learned online. Encoder and decoder accumulate identical histograms from decoded blocks and advance the same model state. The model enters bit allocation directly: per-band rate–distortion frontiers combine measured perceptual distortion with entropy-coded rate estimates under the current model, and a dynamic program attains the exact minimum of total objective perceptual distortion for each block's budget under the model's rate estimates, with reservoir carry-over. Under a strict per-block rate, entropy-coding-aware bit allocation lowers this distortion (band noise-to-mask ratio in linear scale) by 15-24% relative to the same solver without it, and co-adaptation removes a further 3-4% (up to 15% on oboe) that a backward-adaptive coder alone cannot realize in one pass; a MUSHRA test at 80 kbps per channel confirms the ranking.

We propose a continuous learning formulation for $T$ non-forgetting tasks with the spherical parameter space $\mathbb{S}^{d-1}(\sqrt{d})$ in asymptotic regime $d\rightarrow \infty$. Each task $i$ has an optimum $\bm{\Theta}_i$, minimizing its corresponding loss function $R_i(\bm{w})$ on sphere. We reduce each task $i$ to an acceptable loss region modeled by a geodesic ball $B_{\bm{\Theta}_i}(r_i(d))$ on $\mathbb{S}^{d-1}(\sqrt{d})$ centered at $\bm{\Theta}_i$ with radius $r_i(d)$. To model task dependencies (similarities), task optima $\bm{\Theta}_i$ are distributed by a Neyman-Scott clustered point process $\Phi$. Cluster centers $\{\bm{Y}_j\}_{j=1}^{N_p}$ form a homogeneous Poisson point process on sphere, while intra-cluster task optima follow a von Mises-Fisher distribution with mean $\bm{Y}_j$ and concentration $\kappa$. Each task $t\in \{1, ..., T\}$ learning algorithm is considered to be a channel $Q_{t}(\mathrm{d}\bm{w}_t | \bm{w}_{t-1}, \mathcal{K}_{t-1}(\bm{\theta}_{1:t-1}), \bm{\theta}_t)$, where $\bm{W}_t$ is the learned parameter of task $t$ and $\mathcal{K}_{t-1}(\bm{\Theta}_{1:t-1})=\bigcap_{i=1}^{t-1}B_{\bm{\Theta}_i}(r_i(d))$. Since in continual learning, the learner does not need to reach $\bm{\Theta}_t$ exactly to successfully learn task $t$, to avoid forgetting as well as reliable learning, it only needs for the learned parameter vector $\bm{W}_{t}$ to enter $B_{\bm{\Theta}_t}(r_t(d)) \cap \mathcal{K}_{t-1}(\bm{\Theta}_{1:t-1})$. Using this model, we derive an upper bound on $T$ ensuring $\mathbb{P}[\mathcal{K}_{T}(\bm{\Theta}_{1:T}) \neq \emptyset] \rightarrow 1$ as $d\rightarrow \infty$ and $\kappa/d \rightarrow \alpha$, where $\alpha>0$ is a constant. Finally, for $T$ tasks sampled uniformly from $\Phi$, we derive an upper bound for total information theoretic capacity under the non-forgetting constraint for each task.

Learning symmetries through data augmentation offers an alternative to architectural equivariance, but does not guarantee that a network respects those symmetries. Measuring learned equivariance is especially difficult for intermediate representations, whose group actions are not specified. In a finite discrete setting, we show that equivariance requires representations of original and transformed inputs to determine one another exactly, yielding a test based on normalized mutual information (NMI). For high-dimensional activations, we use centered kernel alignment (CKA) as a scalable test, with an exact guarantee for orthogonal feature transformations. Across architectures and tasks, augmentation usually increases transformation consistency, showing that learned equivariance can be probed directly from representations without recovering the hidden transformation rules that relate them.

Recent frameworks in differential privacy (DP), such as privacy loss distributions and $f$-DP, provide lossless representations of the privacy leakage of mechanisms. These approaches are often considered to yield tighter $(\varepsilon, \delta)$-DP guarantees than those obtained from relaxed notions such as Rényi DP. But is that always the case? We study this question for differentially private stochastic gradient descent (DP-SGD) with fixed-size batches under replace adjacency. Our key observation is that in this setting, privacy accounting based on Rényi DP can use a pair of worst-case distributions (i.e., a dominating pair) that differs from the one used in the lossless frameworks. In particular, we characterize the worst-case distributions as two-dimensional Gaussian mixtures and derive an improved Rényi DP bound for DP-SGD. Numerical and asymptotic comparisons show that across various hyperparameter settings, a Rényi DP accountant based on the new bound can yield better privacy guarantees than those obtained from the lossless frameworks.

We study privacy-preserving algorithms that redact information from a correlated dataset. The goal is to reduce the information leakage about sensitive parts. We analytically characterize the optimal tradeoff between privacy (leakage through correlation) and utility (number of redactions) for local mechanisms and binary Markovian correlations. Our result shows how to decompose the mechanism design problem into a sequence of local mechanism design problems and reveals their sequential relation. The optimal mechanism is the solution to a Finite-Horizon Deterministic Markov Decision Process.

Let $q$ be a prime power. An $(r, i)$-regular fat $\mathbb F_q$-linear set has exactly $r$ points of weight $i\ge2$, called fat points. All of its remaining points have weight one, and at least one such point occurs. We solve two open problems posed by three of the authors of this paper. First, we derive divisibility constraints on the point weights using the scalar stabilizer of the defining subspace. Consequently, a regular fat linear set of rank $2i$ in $\operatorname {PG}(1,q^n)$ has at most two or exactly $q + 1$ fat points. In the latter case, the fat points form an $\mathbb F_q$-subline. Second, we construct higher-dimensional regular fat linear sets whose fat points do not form an $\mathbb F_q$-subgeometry. In particular, for every $k\geq 3$ we obtain examples of rank $2k$ whose fat locus consists of an $\mathbb F_q$-subgeometry $\operatorname {PG}(k-2,q)$ in $\operatorname {PG}(k-1,q^4)$ together with one external point. This construction is then generalized further. Finally, we apply our projective-line results to $q$-matroid representability and present an infinite family of non-representable rank-two $q$-matroids arising from partial spreads.

This paper investigates a shift-sum decoding algorithm that employs dual codewords. Existing dual codeword shift-sum decoding schemes mainly rely on minimum-weight dual codewords. However, the number of such codewords is often limited, which restricts the amount of reliability information available to the decoder. To address this issue, we extend the component counting mechanism from minimum-weight dual codewords to dual codewords whose Hamming weights lie within a prescribed upper bound. We further introduce a normalized component counting rule to suppress the overcounting bias caused by the much larger populations of higher-weight classes. Based on this framework, we develop a hard-decision iterative error reduction decoder and a soft information-aided iterative error reduction decoder. Numerical results show that the proposed method consistently outperforms existing minimum-weight shift-sum baselines especially for short codes where additional dual codeword information is particularly valuable.

Matching one distribution to many targets can require repeated evaluations of every target. CAL-$\alpha$-MODM aggregates a nonlinear predictor into one barycenter, calibrates its residual derivatives, and checks steps against the original $f$-divergence objective. Under smoothness assumptions, calibration controls parameter-step model error without requiring targets to be close to the model. Exact-oracle stationarity and local convergence guarantees are established. At common matching, canonical geometry cancels the leading quadratic iteration error, giving cubic convergence with second-order calibration and sufficiently accurate inner solves. High-precision experiments support this prediction. Sensitivity-based initialization also accelerates preference continuation along regular stationary branches. On 300-target Jensen–Shannon problems, CAL-p2 achieves median paired speedups of $3.15\times$ over L-BFGS and $1.10\times$ over second-order Direct-TR, across 23/25 and 24/25 objective-comparable stationary pairs. The savings arise when cheaper predictor queries offset calibration overhead.

We study the information-theoretic limits of inference in rectangular rank-one spiked matrix models with additive rotationally invariant noise, thereby going beyond the standard assumption of independent noise entries. Using the replica method from statistical physics, we derive a variational formula for the mutual information between the signal and the data, together with a system of saddle point equations predicting the minimum mean squared error in estimating the rank-one spike. We further show that, in the Bayes-optimal setting, these saddle point equations coincide with the fixed point equations of the state evolution of a recently proposed orthogonal approximate message passing algorithm with Bayes-optimal denoisers. This equivalence provides an algorithmic interpretation of the replica prediction and supports the Bayes-optimality of orthogonal approximate message passing whenever the relevant fixed point is unique. Numerical evaluations of the saddle point equations for several rotationally invariant noise ensembles and signal priors, together with finite-size simulations of the algorithm, confirm the asymptotic predictions. They illustrate both regimes with a unique informative fixed point, where orthogonal approximate message passing is predicted to attain the minimum mean squared error, and regimes with multiple fixed points, where a statistical-to-computational gap exists.

Existing deep-learning-based channel state information (CSI) feedback methods reduce feedback overhead by compressing high-dimensional CSI into low-dimensional representations and/or exploiting historical CSI to predict future CSI. However, these methods may become unreliable when user equipment (UE) suddenly changes its motion. To address this issue, we propose a prediction-aided coding (PAC) framework for generative CSI feedback, which adaptively encodes CSI based on predicted channel/CSI distributions and the observed CSI in real time. Each CSI frame is represented by multiple CSI tokens obtained without using historical CSI, making token generation robust to abrupt UE motion. In the first stage, the conditional distribution, instead of a deterministic value, of the next-frame CSI tokens is predicted based on historical tokens. In the second stage, once the next frame arrives, the predicted distribution is used for variable-length entropy coding of its CSI tokens. To further reduce feedback overhead and inference latency, we propose efficient designs for both stages. For the first stage, a customized causal Transformer with a new positional embedding strategy accurately predicts next-frame token probabilities. For the second stage, a parallel causal decoding strategy reduces the inference latency caused by sequential token decoding. Extensive experiments show that, compared with existing history-aware baselines, PAC reduces feedback overhead by approximately 72% to 84% while achieving higher reconstruction accuracy. Additionally, PAC maintains CSI reconstruction accuracy under random-walk trajectories, whereas existing history-aware baselines suffer sharp degradation in reconstruction accuracy.

Digital Twin (DT) combined with reinforcement learning (RL) can support adaptive resource allocation and security-aware decision-making by comparing expected and observed network behavior. This paper presents a simulation-based DT reference and a belief-guided Q-learning (BGQL) for joint jammer identification and beam-power control in a site-specific 28-GHz network. For each action, the same channel simulator is evaluated under an attack-free and an attacked configuration. Receiver-level residuals are combined with spatial jamming footprints and accumulated to form a normalized suspicion vector. BGQL incorporates this vector into an aggregated state, a belief-concentration reward, and exploratory beam-selection rule. The BGQL scheme is compared with SARSA, Greedy, and Random under matched attacker sets. Across 30 paired Monte Carlo runs in four configurations, with the number of active jammers supplied to every method, BGQL achieves mean aggregate full-band link-rate scores of approximately 437.4–564.2~Mbit/s and rate-to-device-power scores of 60.37–76.79~Mbit/J. Its detection accuracy is 100% in one-jammer configurations and approximately 78%–81% in two-jammer configurations. BGQL requires approximately 80–130~s per run, compared with 5–14~s for SARSA, reflecting evaluation of 625 beam pairs during exploratory steps. These findings provide a matched-model simulation proof of concept for DT-referenced belief-guided learning for security-aware beam-power control in 6G networks.

This paper considers a distributed estimation system where edge nodes transmit averaged features to a center node for parameter estimation. A key challenge arises when the center node possesses auxiliary side information: how to design local extractors that are universally optimal for any downstream task, while avoiding redundancy with the side information. To tackle this problem, we orthogonally decompose the joint function space and demonstrate in the weak-dependence regime that the modal decomposition of the canonical dependence kernel's conditional-additive component yields optimal, task-independent features. We construct a distributed nested H-score to learn this decomposition and show that, under a limited feature budget, the mean-squared error (MSE) gap to the best task-specific features is determined solely by the task's alignment with the discarded, least informative modes, and vanishes to leading order for every square-integrable task once the budget is sufficient.

The introduction of Reconfigurable Intelligent Surfaces (RIS) technology marks a transformative era in modern communication systems, enhancing coverage and facilitating higher data transmission rates. While recent research has delved into the integration of RIS into satellite networks, a critical and practical gap persists concerning its implementation in the satellite-Very Small Aperture Terminal (VSAT) downlink, specifically amplifying contributions from the VSAT side. This paper addresses this gap by proposing a simple and engineering methodology to optimize RIS position and orientation, incorporating considerations for the antennas' radiation patterns and the system's topology to streamline optimization variables. The primary objective of this paper is to alleviate the double fading constraint in the reflect link without the need for active elements, achieved by leveraging the Fresnel zone of the RIS. The improvement power ratio depends on the frequency and the VSAT 's diameter. The results show that for a VSAT with diameter 1m the gain is up to 9dB at 2 GHz (S band) and up to 14dB at 4 GHz (C band). Additionally, the RIS channel equation is refined, encompassing the tropospheric attenuation and accounting for electromagnetic wave polarization.

The majorization simplex $\mathcal{P}_n$ is the set of probability vectors of length $n$ whose entries are arranged in non-increasing order. Under the majorization order, $\mathcal{P}_n$ is a lattice, and it is the natural space for comparing distributions when the labels of the outcomes are not relevant. In this work, we study measures of dissimilarity (and, similarity) on $\mathcal{P}_n$. Our first set of results concerns the Rényi-entropy based distance $d_\alpha(p,q)=H_\alpha(p)+H_\alpha(q)-2H_\alpha(p\vee q)$ proposed by Yadav and Shkel (2026), which generalizes a metric of Cicalese, Gargano and Vaccaro (2013). We show that $d_\alpha$ is a metric on $\mathcal{P}_n$ for $\alpha\in[1,\infty)$, while for $\alpha\in[0,1)\cup\{\infty\}$, $d_\alpha$ is not a metric in general. Our main result in this part identifies the necessary and sufficient conditions for equality in the triangle inequality of $d_\alpha$ when $\alpha>1$, for which we identify the equality conditions for the supermodularity of the Rényi entropy $H_\alpha$. Our second set of results concerns the total variation distance and the Hellinger affinity (also called the Bhattacharyya coefficient) on $\mathcal{P}_n$. We obtain a number of bounds on these measures, along with some of their equality conditions. The resulting Hellinger affinity based upper bound on the total variation distance can be smaller than the bounds obtained from Pinsker's and the Bretagnolle–Huber inequalities.

This paper delves into the intricate aspects of Reconfigurable Intelligent Surfaces (RIS) in the antenna design. It investigates the properties of the array factor of RIS, highlighting the significance of phase shifts in establishing of the reflect path. Additionally, the paper investigates spatial positions requiring identical phase shifts, nullification of the array factor, the count of total maxima, second maximum determination (amplitude of the second lobe), and the calculation of the 3dB bandwidth. They are very useful for RIS-assisted radio communication systems.

The deep hole problem is a fundamental problem in coding theory, and it has many important applications in code constructions and cryptography. The deep hole problem of Reed-Solomon codes has gained a lot of attention. As a generalization of Reed-Solomon codes under Hamming metric, we study the covering radius and deep holes of Hyperderivative Reed-Solomon (HRS) codes under the Niederreiter-Rosenbloom-Tsfasman (NRT) metric. We determine the NRT covering radius of HRS code and characterize its NRT deep holes via Maximum Distance Separable (MDS) extensions and polynomial remainder conditions. We further relate the enumeration of NRT deep holes to the characteristic polynomial of an associated hyperplane arrangement. We also provide three classes of explicit NRT deep-hole constructions, propose a Cheng-Murray type conjecture for HRS codes under the NRT metric and obtain a conditional low-layer description. Using these characterizations, we derive exact enumeration formulas in several special cases.

Adaptive modulation and coding (AMC) improves the spectral efficiency of underwater acoustic (UWA) links, but its feedback is inevitably stale: sound propagates slowly, and by the time channel state information (CSI) arrives, the fast time-varying multipath channel has moved on. Predictive link adaptation therefore needs more than a point forecast: it needs a distribution over each candidate action's consequences. We propose PRB-WM, a physics-regularized Beta world model for predictive per-subcarrier link adaptation on real time-varying UWA channels. A persistence-anchored stochastic latent-space dynamics model learns how the channel evolves; a counterfactual outcome head evaluates, for every candidate modulation and coding scheme (MCS), a Beta-binomial distribution of bit errors on shared imagined futures; and a chance-constrained receding-horizon model predictive controller maximizes expected spectral efficiency subject to a bounded probability of violating the bit error rate (BER) target. To the best of our knowledge, PRB-WM is the first world model for UWA communications. In closed-loop replay of six measured sea-trial links from the public UWA Channel Library against a 24-method pool spanning the full methodological spectrum, PRB-WM attains a mean goodput 87% above the strongest baseline, ranks first on all six time-varying links at the longest lookahead, and is the only method whose realized frame risk decreases systematically with lookahead, meeting the BER-violation contract where no baseline does. Ablations confirm that rollout sampling dispersion is essential to the chance constraint and that the physics anchor is necessary for trainability.

State estimation is widely used in networked dynamical systems, but released estimates may inadvertently reveal sensitive information about private state variables. Existing privacy-preserving estimation frameworks often do not explicitly account for adversarial prior knowledge. Motivated by this gap, we study privacy-preserving state estimation for discrete-time linear stochastic systems under prior information exposure using pointwise maximal leakage (PML). We first reveal a privacy anti-optimality property of the Kalman filter: in the absence of additive release noise, the steady-state Kalman filter maximizes PML privacy leakage among the considered class of stable linear estimators. We further establish a nontrivial lower bound on the steady-state estimation error enforced by $(\varepsilon,\delta)$-PML privacy. We then derive sufficient release-noise conditions that guarantee $(\varepsilon,\delta)$-PML privacy, first for the Kalman filter and then for stable linear estimators. Finally, a convex semidefinite program is formulated for a joint design of the estimator gain and release noise covariance, providing a tractable privacy-preserving estimator with an explicit upper bound on the steady-state estimation error. Simulations on a source–load power system demonstrate the proposed privacy–utility tradeoff.

Coded computing mitigates stragglers in distributed computation. While Lagrange coded computing (LCC) enables exact polynomial computation over finite fields, Berrut approximated coded computing (BACC) enables approximate computation of arbitrary, including non-polynomial, functions in the analog domain. In applications such as multi-task learning and inference, multiple functions must be evaluated over different subsets of a common dataset, making single-round computation desirable to reduce communication overhead. An existing LCC approach addresses this requirement but is restricted to polynomial functions over finite fields, leaving single-round multi-function computation in the analog domain an open problem. In this work, we develop a single-round construction for computing multiple arbitrary functions using BACC. We first extend the existing multi-function computation approach in LCC to the BACC setting and derive an approximation-error bound. Building on this, we propose a construction based on Berrut basis functions for computing multiple arbitrary functions and derive its approximation-error bound. We analyze the scaling of both bounds with the number of functions and the number of data points per function, and numerically compare the two constructions over a wide range of arbitrary functions. The results demonstrate that the proposed construction achieves lower relative errors than the existing approach in the tested settings.