Papers

The spiked mixture model (SMM) has been introduced as a probabilistic model that generalizes the single-spike (Wishart) model to a mixture model form. With applications ranging from imaging mass spectrometry in the life sciences to hyperspectral imaging in computer vision, it is crucial to understand under which circumstances its signals can be recovered from noisy measurements. The highly multiplexed nature of these measurement types furthermore necessitates such analysis to hold in highdimensional settings. In this paper, we prove that the extreme eigenvalues of the covariance matrix from the SMM exhibit a phase transition in high-dimensional regimes. We show that this phase transition, and thus signal recovery by extreme eigenvalues, depends on several interacting factors: the correlation between spikes (i.e., how similar in content underlying signals are), the energy parameters (i.e., the absolute strength of each underlying signal), and the mixture probabilities (i.e., how likely it is to encounter each underlying signal). This work provides sharp information-theoretic bounds on the parameters needed to detect one or more spikes from extreme eigenvalues of the SMM covariance matrix, and these guarantees could potentially impact any application of the SMM. Understanding this interplay could serve as a tool for driving experimental design in analytical chemistry and life sciences.

The pinching-antenna system (PASS) uses dielectric particles along a low-loss waveguide as reconfigurable passive radiators. Existing analyses conclude that the in-waveguide attenuation is negligible at low frequencies and millimeter wave bands; we show this fails at terahertz (THz), where realizable waveguide losses are dramatically larger. We develop a unified wideband THz-PASS propagation model integrating in-waveguide attenuation, atmospheric absorption, molecular re-radiation noise, and beam squint. Closed-form results follow: a band-averaged coherence factor; a cluster-center placement satisfying a band-edge SINR equalization condition; an associated placement-inversion threshold; and a proposed Switched-Feed PASS (SF-PASS) architecture in which a centrally located radio-frequency switch routes the signal among multiple waveguide segments, with a closed-form insertion-loss payoff threshold. Numerical evaluation at the best PASS-compatible THz operating point shows that SF-PASS substantially outperforms single-feed PASS in spectral efficiency and is competitive with a large-scale antenna array at much lower hardware costs.

With the development of artificial intelligence (AI) techniques, neural receivers, which apply AI to improve wireless receivers have been developed. However, most existing neural receivers apply deep learning only to the outer receiver while retaining conventional channel decoding for the inner receiver, which prevents joint optimization and makes it difficult to build efficient and unified AI-native receivers. To address this issue, we propose a foundation model (FM)-enabled unified neural receiver, FM-Receiver, that integrates the outer and inner receivers into a single AI-native framework, by leveraging the strong representation capability of FMs. Specifically, we introduce a grouped error correction code Transformer that performs symbol-level channel decoding, enabling seamless integration of the inner and outer receiver. Building on this, we illustrate the proposed FM-Receiver, that directly takes the received signals as input of FM and outputs the recovered transmitted bits. In addition, a three-stage configuration-adaptive pre-training strategy is designed to improve the generalization ability to diverse system configurations and scenarios. Extensive simulations show that the proposed FM-Receiver achieves better performance than baselines across different system configurations. It also demonstrates strong zero-shot generalization to unseen frequency bands and scenarios.

We investigate the capacity of an integrated sensing and communication system operating with orthogonal frequency division multiplexing, where the integrated sidelobe level of the transmit signal is adopted as an input cost. The problem is reduced to the capacity of a complex additive white Gaussian noise channel under power and kurtosis constraints. The sensing-optimal and communication-optimal operating points are characterized by circularly symmetric constant-modulus and complex Gaussian inputs, respectively, while the capacity-achieving input in between has discrete amplitude support with a concentric multi-ring geometry. We propose the superposition of the two extremal inputs as a simple and tunable signaling strategy, whose rate admits an exact expression via successive decoding. Numerical results show that the proposed strategy attains a worst-case gap of approximately 0.043 bits to the capacity.

With the growing demand for satellite sensing and communication, the limited wireless resources are difficult to support multiple satellite systems. Therefore, it is desired to investigate integrated sensing and communication (ISAC) in low Earth orbit (LEO) satellite systems to enable multi-functionality within a single satellite, thereby saving both spectrum and orbital resources. In this paper, a framework for ISAC in LEO satellite systems is established, where a satellite can simultaneously sense multiple targets and serve multiple communication users (CUs) over the same spectrum. Considering the limited onboard energy of satellite, a novel robust beamforming design algorithm is developed with the goal of minimizing total transmit power while satisfying the mean squared error (MSE) requirements for sensing and signal-to-interference-plus-noise ratio (SINR) requirements for communication in presence of channel phase uncertainty which exacerbates the cross-functional interference. According to theoretical analysis, the proposed algorithm for ISAC in LEO satellite systems is effective. Moreover, extensive simulations confirm the superiority of the proposed algorithm over baselines.

Next-generation low Earth orbit (LEO) satellite constellations face critical bottlenecks in spectral efficiency and onboard hardware complexity. To overcome these limitations, this paper introduces a novel architecture enabled by metasurface antennas (MAs) at the LEO satellites. In particular, MAs are metasurface-integrated feed antennas that perform high-precision beamforming directly in the wave domain, thereby effectively mitigating multi-user interference. Based on such an antenna architecture, a weighted sum rate (WSR) maximization problem is formulated by jointly optimizing the scheduling of feed antennas to terrestrial users (TUs) and the passive beamforming of the metasurface for system performance enhancement. To address this mixed-integer nonlinear programming (MINLP) challenge, an alternating optimization (AO)-based joint scheduling and beamforming algorithm is proposed. On the one hand, the proposed algorithm incorporates a polynomial-time minimum-cost maximum-flow (MCMF) method, which is dedicated to the optimal scheduling of feed antennas and TUs. On the other hand, it adopts a weighted minimum mean square error (WMMSE) method integrated with semidefinite relaxation (SDR) technique, which is tailored for metasurface beamforming design. Simulation results confirm the effectiveness of the proposed algorithm for MA-enabled LEO satellite constellation communications.

Let $\mathcal{C}$ be an $[n,k,d]$ linear code over $\F_{q^2}$ with parity-check matrix $H$. The rank $r=\Rank(HH^{\dagger})$ measures how far the Hermitian dual $\mathcal{C}^{\perp_h}$ is from being Hermitian self-orthogonal, and it is the least number of coordinates that must be adjoined to $\mathcal{C}^{\perp_h}$ to make it Hermitian self-orthogonal; equivalently, the Hermitian hull $\mathcal{C}\cap\mathcal{C}^{\perp_h}$ has dimension $(n-k)-r$. In this paper, we develop a structure theory of the Hermitian hull of a cyclic code and apply it to the construction of quantum stabilizer codes. For a cyclic code with defining set $T\subseteq\mathbb{Z}_n$, we show that $r$ is additive over the orbits of the involution $\tau(\nu)=-q\nu$ on the $q^2$-cyclotomic cosets and satisfies a parity constraint. A parameter dictionary exhibits $r$ as a gauge trading one-for-one against the code length, and a complementation duality shows that the complement of $T$ realises the logical-qubit count of the quantum code. The extreme values of $r$ are settled by closed subset-sum criteria, and the $\tau$-invariant defining sets are shown to be the unique locus on which the Hermitian dual is again cyclic. On the distance side, we determine the Hermitian dual distance of an $r$-fold extension of an arbitrary linear code exactly, and we prove that the admissible extension vectors lie in the image of $HH^{\dagger}$, which reduces the search space from $q^{2(n-k)}$ to $q^{2r}$. Applied to cyclic, quasi-cyclic and BCH codes, the construction improves $23$ best-known parameters in Grassl's code tables, one of them provably optimal.

Many recently proposed Calderbank-Shor-Steane (CSS) quantum low-density parity-check (QLDPC) codes have sparse decoding graphs, enabling syndrome-based belief propagation (BP) decoding at low complexity. Their construction, however, often results in properties that impair BP performance, such as short cycles and degeneracy. In this work, we propose a general framework for introducing auxiliary variable nodes (AVNs) and auxiliary check nodes (ACNs) into the decoding graph of CSS codes, compatible with the standard stabilizer measurement framework. This provides an additional degree of freedom in the design of the decoding graph itself and can be used to tackle the aforementioned shortcomings. We show that recently proposed techniques, 4-cycle removal and subcode ensemble decoding, can be interpreted as instances of this framework. For 4-cycle removal, we find that the gains depend strongly on the BP iteration count and check-node message scaling. Building on this framework, we further propose a graph-derived subcode ensemble decoder and demonstrate under circuit-level noise that it substantially reduces the per-round logical error rate compared with BP on the corresponding 4-cycle-free decoding graph.

Guessing random additive noise decoding (GRAND) can efficiently decode any moderately redundant code with near maximum likelihood (ML) performance via noise effect guessing. For binary linear codes, Rowshan and Yuan's Segmented GRAND was the first to show that constrained guessing can reduce guesswork. Although powerful, their approach requires a specific parity-check matrix structure that limits the number of constraints that can be exploited as well as the class of applicable codes. Here we introduce GSegGRAND, a generalization of Segmented GRAND that circumvents its limitations. Built on a novel parity check structure and a transformation that maps codes into this structure, GSegGRAND efficiently incorporates up to log2(n) constraints for a wide range of codes, reducing guesswork by an additional 75% over Segmented GRAND. To leverage that advantage for soft-output decoding, we derive an accurate soft-output (SO) equation for GSegGRAND by extending soft-output GRAND (SOGRAND) to incorporate constrained guessing. Applying this SO to turbo product decoding, GSegGRAND achieves up to 88% guesswork reduction, making it a promising candidate for low-latency decoding in future communication systems.

As wireless visual intelligence evolves from isolated task inference to ordered skill workflows, the communication bottleneck shifts from transmitting a single semantic representation to coordinating reusable skill states under channel constraints. Existing DeepJSCC and prompt-guided visual transmitters usually treat each task as an independent full-token transmission, with limited reuse of execution memory across semantic workflows. This is inefficient for workflows such as Detect, Segment, and Keypoint, where later stages often require only state-relevant semantic updates. To this end, we propose SkillComm, a skill-driven semantic communication framework that uses reusable skill states as shared context for workflow-aware token prioritization and memory-assisted token-grid reconstruction. A shared Skill-Book maps a high-level visual intent into a synchronized executable skill sequence at the transmitter and receiver. Conditioned on this workflow, adaptive token selection exploits cross-step memory to transmit only state-active tokens through joint source-channel coding, while the receiver reconstructs a task-ready token grid by combining decoded tokens with local historical memory. Experiments on the MS COCO 2017 validation set for the Detect-Segment-Keypoint workflow show that SkillComm reduces token transmission cost by 51.2% while retaining 99.4% upper-bound-normalized average precision at high SNR. These results demonstrate that reusable skill states enable selective semantic update delivery for future agentic and embodied visual intelligence.

We develop an enhanced in-context learning (ICL) framework to improve the performance of pilot-based beamforming in multi-user multiple-input single-output (MU-MISO) systems. The proposed scheme integrates the ICL-Transformer backbone with the pilot encoder-decoder network (EDN) and the beamformer EDN. A crucial feature of our ICL network is that it can handle multiple channel models without retraining, enabled by the construction of model-specific context datasets. To improve convergence and robustness, we introduce three key innovations: (a) a curriculum learning (CL) strategy that smoothly transitions from supervised LMMSE-labeled imitation to unsupervised sum-rate maximization, (b) a self-evolving mechanism that dynamically expands and refines the context datasets for all channel models during CL-based training, and (c) a mismatch-aware extension that incorporates several mismatches into the general ICL framework and bypasses explicit channel calibrations. Ablation studies validate the effectiveness of the in-context architecture and enhanced training strategies. Simulation results over diverse communication environments show that the proposed scheme is able to rapidly adapt to both seen and unseen channel models without gradient-based parameter updates, and can mitigate the mismatch issues via intelligent context constructions. Furthermore, our scheme consistently outperforms the existing beamforming schemes under pilot-based settings, including the WMMSE benchmark and the recent Transformer-based methods.

Extremely large-scale multiple-input multiple-output (XL-MIMO) is a promising technology for supporting automated guided vehicle (AGV) fleets in smart port terminals. However, the metallic container environment induces spatial non-stationarity, whereby each AGV is visible to only a subset of the array, referred to as its visibility region (VR). Unlike existing XL-MIMO models that assume user-independent VRs, we show that each AGV simultaneously acts as a communication user, a metallic scatterer, and a blocker, resulting in coupled user channels and VRs. We formulate this triple-role effect through a VR-coupled channel model and develop a VR-aware downlink beamforming framework based on masked weighted minimum mean-square error (WMMSE), where the masking operation exactly enforces VR support constraints while significantly reducing computational complexity. Simulation results in a realistic smart port scenario demonstrate more than a threefold sum-rate improvement over VR-unaware baselines, with the gains becoming increasingly pronounced as fleet density increases.

Character-polynomial codes are constructed by evaluating finite field polynomials and mapping the results to complex roots of unity through additive characters. This paper shows that, over extension fields, the original polynomial family may contain redundancies: distinct polynomials can generate the same codeword. We identify the source of this non-injectivity through the trace map and cyclotomic cosets, determine the exact code cardinality, and construct a refined polynomial family that parametrizes the code without redundancy. These results give corrected parameters for CP codes and clarify their algebraic structure.

Long range-frequency hopping spread spectrum (LR-FHSS) is a promising uplink physical layer for massive low Earth orbit satellite Internet of Things, where low power terminals report short packets from wide area regions with limited terrestrial infrastructure. However, satellite IoT links are exposed to external interference, and the coexistence of multiple interference components can severely degrade receiver reliability and complicate interference mitigation. Existing recognition methods either focus on single interference scenarios or treat each compound interference combination as an independent class, leading to limited generalization or poor scalability. To address this problem, this paper formulates LR-FHSS uplink compound interference recognition as a multi-instance multi-label learning problem and proposes a multi-domain instance fusion method. The proposed method fuses local instances from the time-frequency and frequency domains and aggregates their predictions for bag-level multi-label recognition. A dataset construction pipeline is developed based on the US915 LR-FHSS configuration and incorporates shadowed-Rician fading and time-varying Doppler to emulate practical satellite communication conditions. Considering the difficulty of obtaining labeled compound interference samples in practice, single-to-compound generalization and few-shot compound interference adaptation are investigated as two practical receiver deployment scenarios. Experimental results show that the proposed method improves the overall exact accuracy over the strongest baseline by 14.71 percentage points in single-to-compound generalization and by 14.81 percentage points in few-shot compound interference adaptation for $r=1$.

Low-bit quantization makes small reasoning models inexpensive to deploy but can degrade their chains of thought. This motivates decoder-side monitors that intervene when generation becomes unreliable. We show that a natural candidate, the centered token log-probability increment $\log p(w_t)+H_t$, is the wrong observable for this purpose. Under the model's own sampling law it is a mean-zero martingale by construction, so it measures sampling self-consistency rather than trajectory health and is nearly silent during confident repetition, where both $\log p(w_t)$ and entropy are close to zero. We introduce a training-free decoding controller that combines (i) a degeneration-aware alarm score fusing token uncertainty with explicit verbatim repetition and (ii) a calibrated e-process-inspired sequential detector. The raw product process is Ville-valid under a conditional-mean null, while the deployed CUSUM-floored statistic is treated as an empirical change detector because the score is history-dependent and autocorrelated. On GSM8K with DeepSeek-R1-Distill-Qwen-1.5B in FP16 and INT4, calibration turns a monitor that fires on 93–95% of generations into a selective detector of failing traces ($\phi \approx 0.3$, precision $\approx 0.6$ against a 0.38 base rate). In this pilot, the controller reduces measured verbatim-degeneration signals and yields a positive but statistically inconclusive INT4 accuracy change from 63% to 69% (paired McNemar $p=0.18$, $n=100$), at a 28% token-budget cost. We also find that non-termination, rather than looping, is the dominant failure mode on GSM8K. The main contribution is methodological: an explanation of why centered token log-probability is inadequate for decoder monitoring and a calibrated, cautiously evaluated replacement.

Joint communication and sensing has been a key technology in 6G. By integrating sensing into maritime communications, ships can communicate with the base station while sensing the surrounding environment to ensure safe navigation. In this paper, we introduce an integrated satellite-terrestrial maritime system (ISTMS) with joint communication and sensing based on the same radio-frequency signals. Specifically, the terrestrial base station (TBS) and low Earth orbit (LEO) satellite provide communication services for near-shore users (NSUs) and off-shore users (OSUs), respectively, while simultaneously performing target sensing. Based on a differential evolution method (DE), we propose a sensing algorithm, which can enhance the location accuracy and reduce resource consumption. Furthermore, we derive the key performance metrics for both communication and sensing. Through joint beamforming optimization of the TBS and LEO satellite, we maximize the sum rate of maritime users while satisfying target localization accuracy requirements and transmit power constraints. Finally, extensive simulation results demonstrate the effectiveness of the proposed algorithms in terms of location accuracy and transmission rate compared with the baseline algorithms.

In this paper, we present an integrated terrestrial and satellite maritime communication system, where a shorebased terrestrial base station (TBS) and a low Earth orbit (LEO) satellite cooperatively provide wide-area communication services to maritime users. We conduct performance analysis for the integrated satellite-terrestrial maritime communication system. Specifically, we analyze the transmission rate and coverage probability of near-shore and off-shore users respectively according to the maritime communication environment. Besides, in order to better understand the impact of some key parameters, we also make asymptotic analysis in some special cases. Further, we design an optimization algorithm to maximize the coverage probability of near-shore users by adjusting the transmission power of TBS and the LEO satellite, while ensuring the both off-shore and near-shore users can meet the minimum communication rate requirements. Finally, extensive numerical analysis results verify the accuracy of the theoretical results and the effectiveness of the proposed optimization algorithm in the maritime communication system.

We study distributed experimental design with multiple candidate experiments, where local sites possess only partial information and transmit design recommendations to a fusion center. Unlike centralized design, in which the experiment that maximizes expected information gain can be selected directly, distributed design requires combining heterogeneous and potentially conflicting local recommendations. Formulating as a multi-class Bayes fusion problem, centralized oracle design is treated as an unknown label and each site is characterized by a local recommendation mechanism. The proposed fusion rule minimizes posterior expected information regret, rather than merely maximizing the number of local votes or the posterior probability (MAP) of the oracle label. This distinction is essential because different incorrect experimental choices may incur different losses in information gain. We show that majority vote is optimal only under restrictive symmetry assumptions and can otherwise be strictly suboptimal. Regret-weighted multi-class Chernoff bounds are derived to identify the pairwise separations governing distributed design performance. Numerical studies identify two operational regimes: MAP is effective when oracle-label accuracy and information regret are aligned, while regret-weighted Bayes fusion reduces information loss when the most probable oracle label is not the lowest-regret decision.

Holographic multiple-input multiple-output (HMIMO) has emerged as a promising technology for future wireless systems by enabling continuous electromagnetic (EM) field control over large apertures. However, user-side EM exposure has become an increasingly important concern in large-scale array systems. This paper addresses this issue by developing a multiuser uplink HMIMO model, where a physically consistent specific absorption rate (SAR) model is established to quantify the EM exposure. On this basis, a spectral efficiency (SE) maximization problem is addressed by developing a modified iterative water-filling algorithm. Simulation results demonstrate that the proposed algorithm effectively improves the system SE while satisfying the SAR constraints.

Let $I_2$ be the commutative non-unitary ring of order $4$ arising in the classification of Fine. In this paper, we investigate cyclic codes over $I_2$ through their associated residue and torsion codes over $\mathbb{F}_2$. We introduce the notions of twisted and untwisted cyclic codes and characterize cyclicity in terms of a compatibility condition involving the twist map and the cyclic shift. Connections between cyclic codes over $I_2$ and binary quasi-cyclic codes are established via Gray maps. In particular, we show that the Gray image of a cyclic code over $I_2$ is a binary quasi-cyclic code of index $2$. We also study duality properties of cyclic codes over $I_2$ and prove that the dual of a cyclic code is again cyclic. Finally, we classify permutation inequivalent cyclic codes over $I_2$ for lengths $n \le 7$ and determine various structural properties of these codes.

We study the problem of releasing counting-query outputs through a stochastic mechanism that is both consistent and \((\epsilon,\delta)\)-differentially private. Consistency requires the released value to lie within the feasible range of the query, while utility is measured by the worst-case probability of error. We first derive a closed-form expression for the minimum achievable error probability and obtain an explicit optimal mechanism. By exploiting the active differential privacy constraints satisfied by this mechanism, we then characterize the entire class of optimal mechanisms via a propagation argument, identifying the structural properties shared by all optimizers. We next extend the framework to the setting in which the privacy mechanism is cascaded with an arbitrary fixed stochastic transformation representing a predetermined portion of the communication medium between the source and the destination. We first establish necessary and sufficient conditions under which this partial fixation of the medium incurs no loss in utility. We then derive upper and lower bounds on the optimal achievable performance based on convex mixing and spectral perturbation. Finally, we apply the theory to (M)-ary phase-shift keying (PSK) transmission over an additive white Gaussian noise (AWGN) channel and show that uncoded transmission is effectively optimal in the high-privacy regime.

I develop a Hamiltonian framework for ${\mathbb Z}_k$ $p$-form gauge fields on arbitrary oriented cell complexes in arbitrary dimensions. Gauge qudits are defined by $p$-cells, charged boundary qudits by $(p-1)$-cells, Gauss-law generators by boundary map $\partial_p$, and magnetic checks by $\partial_{p+1}$. The same cellular structure produces local dressed Wilson operators, and at $k=2$ a Calderbank-Shor-Steane check complex relevant to quantum error correction. I then specialize to $p=2, k=2$, where the magnetic 3-cell term is absent and the one-form Gauss-law can be solved exactly. The physical Hilbert space is parameterized by plaquette electric-flux variables, while the link configuration is reconstructed as the dynamical boundary of the evolving flux domains. The reduced Hamiltonian is an Ising-type plaquette model, where its local transverse-field term is the physical image of the boundary-dressed Wilson operator $\sigma_p^z\prod_{\ell\in\partial p}\tau_\ell^z$. A tube-cap quench compares two initial flux fillings with the same initial boundary loops. Exact diagonalization on $4\times4$, $6\times4$, and $5\times5$ tori finds that the cap loses $20$-$37\%$ of its occupied-flux area, while the tube remains nearly pinned. A finite-size scaling locates a dynamical crossover of tension-to-density ratio near $(m/\varepsilon_E)_c\simeq1.89$. The unreduced plaquette-plus-link encoding provides local Gauss-law checks and a direct digital implementation, while the reduced plaquette-only Hamiltonian supplies the exact benchmark. The result places the specific top-form discharge and the cosmological constant neutralization calculation inside a general higher-form Hamiltonian and coding framework.

The dual lattices $E_6^*$ and $E_7^*$ are of particular interest in source coding and data compression applications. Among all known lattices in dimensions six and seven they attain the smallest normalized second moments, i.e., the smallest average quantization error. Their use in practice requires fast closest-point (nearest-lattice-point) algorithms. The known approach, due to Conway and Sloane and completed for $E_6$ and $E_6^*$ by Takizawa, Yagi, and Kawabata (TYK), decodes these lattices as unions of cosets of root lattices $A_n$: each coset is decoded separately, and the best result is kept. This requires four coset decodings for $E_7^*$ and six for $E_6^*$, together with explicit distance computations. This paper shows that all these coset decodings can be collapsed into a single sweep. Reformulated in terms of glue vectors, the TYK decompositions state that $E_7^*$ is the union of the even glue classes of $A_7^*$, and that $E_6^*$ is a parity-matched sublattice of $A_1^*\oplus A_5^*$. The candidate chain constructed by the closest-point algorithm of McKilliam, Clarkson, and Quinn (MCQ) for $A_n^*$ visits every glue class of $A_n$ exactly once and is optimal within each class. Consequently, one sorted sweep per coordinate block yields the closest points of all glue cosets simultaneously, and $E_6^*$ and $E_7^*$ are decoded at roughly the cost of a single $A_5^*$ or $A_7^*$ quantization. Rough operation counts indicate a $4$–$6\times$ reduction for $E_6^*$ and $3$–$4\times$ for $E_7^*$ relative to coset-by-coset decoding. We also discuss further constant-factor improvements available from recent refinements of the $A_n^*$ algorithms, and an open question concerning sort-free linear-time decoding.

Codes over non-unitary rings have been studied recently. In particular, codes over the commutative non-unitary ring $I_p$ (in the classification of Fine) of order $p^2$ where $p$ is a prime are being considered. For $p=2$ (resp. $p=3$), three categories of codes over $I_p$ have been studied: self-orthogonal codes, quasi self-dual codes, and self-dual codes over $I_p$. Using some related mass formulas and building-up constructions, classifications of these codes have been done up to the permutation equivalence (resp. the monomial equivalence) for certain small lengths. In this paper, we take the prime $p=5$ and consider the ring $I_5$. We introduce the notion of linear codes over $I_5$. We also define the same three categories of linear $I_5$-codes, study the structures of these $I_5$-codes and relate them to their associated residue and torsion codes. We classify the three categories of codes completely in lengths at most $4$ up to the monomial equivalence for a given type $\{ k_1 , k_2 \}$. Moreover, in the paper of Alahmadi et al. regarding the mass formula for self-orthogonal codes over $I_p$, mistakes in the classification of quasi self-dual codes over $I_5$ had been made such as incorrect automorphism group order of some codes or inconsistency with the mass formula for self-orthogonal codes over $I_p$ for length $n=2$ and type $\{ 1 , 0 \}$ and for length $n=3$ and type $\{ 1, 1 \}$. We correct and improve such results.

A binary Coxeter code associated with a finite Coxeter system $(W,S)$ is an ${\mathbb F}_2$-linear span of indicators of standard cosets of a fixed rank. Coxeter codes, introduced in a recent paper by N. Coble and A. Barg, are a generalization of Reed–Muller codes which arise when $W={\mathbb Z}_2^m$ is the Coxeter group of type $mA_1$. In that paper, the authors proposed a conjectural value for the minimum distance of a general Coxeter code. This conjecture is proved in the present work. As a consequence, we obtain a Coxeter-theoretic generalization of Reed's majority-logic decoding algorithm for Reed–Muller codes.

We study sparse-array near-field beam focusing with spatial interference suppression, a problem arising in coherent satellite formations and other distributed non-terrestrial arrays. State-of-the-art designs solve it numerically through second-order cone programming (SOCP) with cutting-plane refinement, yet the achievable signal-to-interference ratio (SIR) and its link to classical adaptive beamforming have remained without an analytical characterization. We supply this characterization via a Lagrangian-dual analysis, obtaining three results. First, every optimal beamformer is a generalized matched filter against an effective spatial covariance induced by an optimal dual measure; this closed form recovers MVDR, LCMV, and SOCP-based focusing as special cases. Second, the dual measure has finite support of cardinality at most $M^2$ in general, sharpening to $M$ for uniform linear arrays ($M$ the number of array elements), which yields a finite-dimensional convergence certificate for cutting-plane methods. Third, a closed-form upper bound on the mean-SIR admits an asymptotic logarithmic scaling law in $M$ under near-collinear geometry, identifying array order, rather than the optimization algorithm, as the dominant performance factor. A Riemannian conjugate-gradient algorithm on the unit-torus manifold is developed for practical constant-modulus beamforming, and numerical results demonstrate that it closely approaches the derived performance limit.

We obtain a Wei-type duality between the footprint bound and the dual footprint bound for the generalized Hamming weights of an evaluation code. This duality applies between the Andersen-Geil and Feng-Rao bounds as well. We also prove that the footprint and dual footprint bounds cannot be used to guarantee the asymptotic goodness of a family of evaluation codes.

In contrast to most studies on neural network approximation theory that characterize results through a single parameter, such as the total number of network parameters, \cite{shen2020deep} pioneered the characterization of approximation rates as a joint function of the width parameter $N$ and the depth parameter $L$, thereby granting greater architectural flexibility. Existing works using the $(N,L)$-characterization focus on function classes with finite smoothness $s$, establishing a typical approximation rate of $\mathcal{O}\left(N^{-2s/d}L^{-2s/d}\right)$ with $d$ denoting the input dimension, which indicates that network depth and width play symmetric roles for these classes. In contrast, this paper establishes upper bounds for the approximation of analytic functions, which possess infinite smoothness, via ReLU networks under the $(N,L)$-characterization. Specifically, we derive approximation rates of $\mathcal{O}\left(N^{-C L^{\tau}}\right)$, where $C>0$ is some constant and $\tau>0$ is a parameter influenced by the relation between $L$ and $N$. In particular, $\tau=1$ if $N$ scales roughly as $L^d$. Our findings reveal that depth plays a more critical role than width in the context of analytic function approximation. The main technical difficulty of obtaining such upper bounds lies in the trade-off between the smoothness parameters and the approximation accuracy. To overcome this difficulty, we employ refined constructions of several ReLU networks to approximate power functions, multivariate multiplication, and polynomials, which may be of independent interest.

Mutual correlated agreement captures whether a random linear combination of received words can create a new large agreement with a code, a property relevant to the soundness of batched proximity testing. We show constructively that list-decoding counterexamples yield lower bounds on the mutual correlated agreement error. Given an explicit counterexample to the $(p,L)$-list-decodability of a linear code over $\mathbb{F}_q$, we construct a related code $C'$ of the same length and dimension such that $\operatorname{err}_{\mathrm{MCA}}(C',p)\ge\frac{1}{q}\left\lceil\frac{(L+1)q}{q+L}\right\rceil$, while decreasing its minimum distance by at most one. The construction also produces an explicit pair of words witnessing this error. We further give a structure-preserving version for code families whose coordinates are indexed by a finite set $\Omega$, with each index determining a generator-matrix column through a map $v:\Omega\to\mathbb{F}_q^k$. The construction changes at most one coordinate index and ensures that the output code remains in the same indexed family. As applications, we instantiate this principle for algebraic-geometry (AG) evaluation codes and Reed–Solomon codes. For AG codes, if $G$ is the divisor defining the underlying Riemann–Roch space and $N$ is the number of rational places outside $\operatorname{supp}(G)$ available for evaluation, the resulting code remains over the same function field and Riemann–Roch space, with a modified set of evaluation places. Its mutual correlated agreement error is at least $\frac{1}{q}\left\lceil\frac{(L+1)N}{N+L\mathrm{deg} G}\right\rceil$. The Reed–Solomon conclusion follows as the Vandermonde-column specialization.

Sparsity is inherently exploited in sparse code multiple access (SCMA) and sparse vector coding (SVC), yet the interaction between these two has not been explored before. It is intriguing to ask if one can be used to improve the other, and vice versa. In this work, we present a novel SCMA inspired SVC scheme, called SCMA-SVC, for enhanced ultra-reliable low-latency communications. Our key idea is to exploit the sparse pattern and multidimensional constellation nature of SCMA, with which one is able to further enlarge the minimum Euclidean distance (MED) of the corresponding SVC codebooks. Such an innovation allows us to harvest the multiuser coding gain and the constellation shaping gain which are pertinent to SCMA. Moreover, by applying random phase rotations to the sparse vectors, it is shown that the proposed SCMA-SVC achieves full diversity order over Rayleigh fading channels. Under maximum likelihood (ML) decoding, the proposed SCMA-SVC demonstrates remarkable error rate performances over both Gaussian and Rayleigh fading channels. Additionally, we develop a low-complexity decoder that exploits the structural sparsity of SCMA-SVC while maintaining near-ML performance. Simulation results demonstrate that the proposed SCMA-SVC achieves significantly improved reliability over the existing SVC variants.