Papers

The Etzion-Silberstein conjecture asserts that the Singleton-type upper bound for linear Ferrers-diagram rank-metric codes is attained for every Ferrers diagram, minimum rank distance, and finite field. Let $E$ be the Ferrers diagram with column heights $(5,5,5,5,1,1)$. The bound for minimum rank distance $3$ is $12$. We prove that every binary linear code supported on $E$ with minimum rank distance $3$ has dimension at most $11$, and we give an explicit code of dimension $11$. Thus the optimum is exactly $11$, disproving the conjecture. The nonexistence proof reduces a hypothetical dimension-$12$ code to one of the three equivalence classes of binary $[4\times 4,12,2]$ MRD codes. A rank-distribution argument eliminates two classes and leaves four kernel orbits in the field class; all four exact lift systems are unsatisfiable. Independently written verifiers reproduce the result, including a raw enumeration of all $8,382,465$ kernels without orbit reduction. We also prove an exact row-cone propagation identity. Iterating it produces binary counterexamples with bound $12$ and optimum $11$ at every minimum rank distance $d \geq 3$.

Throughout the last decade, Gaussian universality has been widely studied for high-dimensional estimation problems. Most of the literature focuses on i.i.d. sensing matrices or accounts for special forms of dependence, such as block dependence or other specific row/column dependencies. More general simultaneous row and column mixing has not yet been fully studied. In this paper, we focus on that setting. We prove a Gaussian universality theorem for the lasso in the sparse regime, where the non- Gaussian covariates have linearly dependent rows and columns. To the best of our knowledge, our setting permits a broader simultaneous row and column dependence structure than those treated in much of the prior universality literature. Numerical illustrations for various sparse profiles support the universality claims of this paper.

Price forecasts are evaluated in EUR/MWh of error, while storage earns euros; their link depends on the decision rule. We separate the optimal value of a signal V(S|C) from the revenue J(g,S) achieved by an implemented policy. For a price-taking asset with daily throughput bound L and a conditionally sub-Gaussian price law of scale s, the optimal gain over climatology is at most L s sqrt(2 I(S;Pi|C)). This is a square-root envelope, not a prediction for an individual policy. On 939 French day-ahead days the tested bound sits at least two orders of magnitude above the reference uplift. Along a nested Gaussian garbling family, however, plug-in policy revenue is non-monotone even though signal information is monotone: at the residual-noise scale the best Gaussian-shrinkage rule achieves -46% of the uplift, below climatology. The mechanism is ordinal. On a relaxation, optimal schedules depend on interval ordering and profitable-pair tests. A constructed rank-only bid captures 90% of the uplift; exchanging the cheapest and dearest entries of an otherwise exact price vector cuts total revenue by 53%. Within the designed sweep, rank agreement has in-sample R^2 = 0.99, against 0.34 for the information upper bound. Climatology already earns 78% of perfect-foresight revenue. Information quantity alone therefore does not value a forecast-driven policy; realised schedules and ordering must be evaluated explicitly.

This paper studies the uplink performance of OFDM-based cell-free massive MIMO systems in the presence of hardware impairments affecting both low-cost access points and wireless fronthaul transceivers. We consider a centralized architecture with functional split option~8, where the sampled baseband signals are forwarded from the access points to the central processing unit through wireless fronthaul links operating at mmWave or sub-THz frequencies. A unified analytical framework is developed to characterize the aggregate impact of hardware distortions across the access and fronthaul links. In particular, closed-form expressions are derived for channel estimation and uplink spectral efficiency under hardware-impaired transmission and reception. To mitigate the resulting performance degradation, distortion-aware linear receive combining is considered for the wireless fronthaul. Moreover, a max-min fair resource allocation framework is developed to jointly optimize the uplink transmit powers, fronthaul transmit powers, and fronthaul time expansion factor.

Analog error-correcting codes (Analog ECCs), introduced by Roth [1], address errors in vector-matrix multiplication arising from analog noise and sparse outliers in in-memory computing. A fundamental open problem concerns the lower bound on the single-error correction threshold $\Gamma_2(\mathcal C)$ for real $[n,k]$ linear codes with fixed redundancy $r=n-k\geq 2$. Li et al. [2] recently established that for redundancy $r=2$, every real $[n,n-2]$ linear code $\mathcal{C}$ satisfies $\Gamma_2(\mathcal C)\geq \csc^2(\frac{\pi}{2n})$, resolving an open problem in [1], and showed that, for every fixed $r \geq 2$, there exists a class of $[n,k]$ linear code $\mathcal{C}$ over $\mathbb{R}$ such that $\Gamma_2(\mathcal{C}) \leq O(n^{1+\frac{1}{r-1}})$. This paper proves the matching converse in [2]. For every $[n,k]$ linear code $\mathcal{C}\subseteq \mathbb R^n$ with fixed redundancy $2\leq r<n$, we show that \[ \Gamma_2(\mathcal C)\ge \frac{a_r}{\sqrt{r}\,\beta_{r-1}\,2^{\frac{1}{r-1}}}\cdot n^{1+\frac{1}{r-1}}, \] where $a_r =\frac{\Gamma(\frac{r}{2})}{\sqrt{\pi}\,\Gamma(\frac{r+1}{2})}$ and $\beta_d = \left(\frac{d\pi^{d-1}|\mathbb S^d|}{|\mathbb S^{d-1}|}\right)^{\frac{1}{d}}$ for positive integer $d$. Here $\mathbb S^d$ denotes the unit sphere in $\mathbb R^{d+1}$, $|\mathbb S^d|$ its surface area, and $\Gamma(\cdot)$ the Gamma function. In particular, we further show that $\Gamma_2(\mathcal C)\geq \frac{1}{4\pi\sqrt{3}r}\cdot n^{1+\frac{1}{r-1}}$. Together with the upper bound in [2], this confirms that the exponent $n^{1+\frac{1}{r-1}}$ is optimal, completing the asymptotic characterization of the single-error correction threshold for Analog ECCs.

Wireless visual uplinks increasingly carry structured representations for edge inference, making packet reliability part of task-aware link adaptation. In open-vocabulary scene-graph transmission, an indexed triplet is usable only if both its triplet packet and the vocabulary packets defining any newly introduced tokens are recovered. This prerequisite coupling makes the marginal value of packet reliability depend on neighboring packet reliabilities. We formulate a dependency-aware semantic distortion and jointly optimize finite choices of modulation and coding scheme (MCS), transmit power, and Chase-combining hybrid automatic repeat request (HARQ) depth under expected delay and energy constraints. The distortion is multi-affine in packet failure probabilities and cannot, in general, be reduced to static separable unequal error protection (UEP) weights when prerequisites are active. This structure yields a state-dependent reliability coefficient and explicit switching thresholds among wireless actions. A Lagrangian block method performs exact per-packet finite-action updates for fixed multipliers. On reduced instances, it matches exhaustive optimization in 28 of 30 cases, with a worst gap of 1.095%. On GQA traces using a table-driven block error rate (BLER) abstraction, it reduces mean semantic distortion by 58.71% and grounded-query failure by 57.03% relative to dependency-agnostic HARQ under the same budgets.

Level-constraints model one-bit-quantized signaling over real-alphabet continuous-time channels. New lower bounds on the capacity of bandlimited, additive white Gaussian noise channels with level-constrained inputs are derived by using probabilistic temporal shaping (PTS). The optimal shaping density is derived for signals with one data-dependent sign change per Nyquist-rate sample, which ensures they satisfy an invertibility requirement. Calculations show that PTS improves the best existing lower bound by at least 1.94 dB at high signal-to-noise ratio (SNR). A simpler sequential PTS scheme achieves a gain of 1.64 dB at high SNR.

The shape function of a pair of finite-valued random variables was introduced in arXiv:2606.23849, where it was used to derive a spectral bound on the entanglement of the pair, a quantity measuring the extent to which their mutual information can be extracted. In this article, we further develop the theory of shape functions for pairs of random variables. We prove that, when $(X,Y)$ is uniformly supported on the edges of a biregular bipartite graph, the value of the shape function $\mathcal{S}(X,Y)(\alpha,\beta)$ equals the logarithm of the operator norm of the graph's incidence matrix with respect to Lebesgue exponents determined by $(\alpha,\beta)$.This identification, in particular, enables the numerical approximation of the shape function and, by duality, of the extension profile, also known as the tension region, of the pair. We also establish a collection of relations and inequalities satisfied by shape functions, including convexity and monotonicity properties, composition inequalities, and relations describing their behavior under conditioning and the adjoining of variables.

In this paper, we focus on constructions of unique-decodable/list-decodable on the recently studied $(t,e)$-composite-asymmetric error-correcting codes ($(t,e)$-CAECCs). Let $X$ be an $m\times n$ binary matrix, in which each row has Hamming weight $w$. When at most $t$ rows of $X$ suffer from errors and in each of these erroneous rows, there are at most $e$ $1 \to 0$ errors, we say that a $(t,e)$-composite-asymmetric-error occurs in $X$. For general $m,n,w,t,e$, we propose new constructions of $(t,e)$-CAECCs with redundancy at most $(t-1)\log(m)+O(1)$, where $O(1)$ is a number independent of the code-length $m$. In particular, this gives a class of $(2,e)$-CAECCs that are optimal in terms of their redundancy. %(in terms of redundancy, regarded as a function of the number of rows $m$) s. When $m$ is a prime power, the redundancy can be further reduced to $(t-1)\log(m)-O(\log(m))$. To further increase the size of these codes, we introduce a combinatorial object called a weak $B_e$-sets. When $e=w$, we show an efficient way to encode/decode our codes. At last, we investigate how much we can gain if we relax the requirement of uniquely decoding to list-decoding. It is shown that when the list size is $t!$ or an exponential function of $t$, there are list-decodable $(t,e)$-CAECCs with constant redundancy. When the list size is two, we show that there are list-decodable $(3,2)$-CAECCs with redundancy $\log(m)+O(1)$.

We study the achievable rate performance of dual-polarized passive reconfigurable intelligent surfaces (RISs) and dual-polarized network-controlled repeaters (NCRs) in a point-to-point wireless link. While RISs rely on passive beamforming with large array gains, NCRs exploit active amplification across polarization branches. We develop a unified analytical framework that captures the impact of amplification, noise propagation, and deployment geometry under different power-budget models. Closed-form rate expressions are derived, and the optimal one-stream and two-stream transmission strategies for NCRs are characterized. The results reveal that RISs are competitive in short-range scenarios and benefit from large array sizes, whereas NCRs become preferable in long-range deployments due to their ability to compensate for pathloss through amplification.

Data storage in DNA has recently emerged as a promising archival solution, offering space-efficient and long-lasting digital storage. Combinatorial DNA encoding enhances this potential by increasing the logical density through combinations of DNA shortmers, where each sequence position is represented by a set of predefined short DNA fragments, allowing more data to be encoded using fewer synthesis cycles. However, this method introduces unique synthesis and sequencing errors. In this study, we characterize errors in combinatorial DNA-based storage systems. We reveal that asymmetric combinatorial erasure errors, defined as the omission of a single shortmer from the set defining the combinatorial letter, are a prevalent error type, particularly in large-scale systems where read coverage is limited. In two previously published datasets, we observed a high frequency of erasure errors, where missing sequences obstruct the reconstruction of combinatorial letters. We conducted a large-scale experimental proof-of-concept and confirmed that erasure errors become increasingly prominent with reduced sequencing depth: below 50 reads per sequence, their frequency sharply increased. We developed an asymmetric error-correcting code for these errors, utilizing tensor-product codes to integrate standard erasure and substitution-correcting codes (such as Reed-Solomon (RS) codes) with asymmetric Varshamov-Tenengolts codes. We validated its performance in simulations and in a second large-scale experiment directly comparing it with the more straightforward 2D RS scheme. Our method consistently outperformed 2D RS, particularly in erasure-dominated scenarios, and demonstrated superior decoding accuracy under low coverage conditions where 2D RS struggled to decode the data. Our findings demonstrate the importance of error correction schemes tailored to the asymmetric nature of errors in combinatorial DNA.

Cell-free massive MIMO with wireless fronthaul is a promising architecture for energy-efficient 6G networks, but the access and fronthaul links must then share the same scarce spectrum, and, under the fully centralized (option-8) functional split, the fronthaul rate is dictated by the finite quantization resolution used at the access points (APs). This paper develops a network energy-efficiency (EE) maximization framework for the uplink of such a system, jointly optimizing the integrated access and fronthaul (IAF) resource split, the adaptive per-AP quantization resolution, and the fronthaul powers, and treating the time-division (TD) and frequency-division (FD) operating modes in a unified manner. Each AP may be switched off (put to sleep) when it is not worth activating, so the resolution allocation is inherently coupled with AP selection. The resulting mixed-integer, nonconvex fractional program is solved by an alternating-optimization algorithm with per-block optimality guarantees—a closed-form optimal time split, bandwidth bisection, and optimal per-AP bit selection—that applies verbatim to both modes. While the design relies on the tractable additive quantization noise model, the reported performance is obtained end-to-end with the actual Lloyd–Max quantizers and a Bussgang decomposition-based achievable-rate bound.

This paper proposes semantic-aided iterative decoding (Sem-IR) for uplink non-orthogonal transmission of a shared natural-language source. K users each hold one segment of a common sentence and superimpose low-density parity-check (LDPC) coded transmissions over an additive white Gaussian noise (AWGN) channel. At the base station, an iterative elementary signal estimator (ESE) and K parallel LDPC decoders progressively cancel inter-user interference. As high-power users pass both parity and language-plausibility checks earlier, their decoded bytes form a reliable linguistic prefix for the remaining users; a fine-tuned ByT5 byte-level language model exploits this prefix to predict byte posteriors for the unconverged user. The byte posteriors are marginalized to bit-level log-likelihood ratios and convex-combined with the LDPC posteriors inside the iterative loop. The resulting feedback closes the loop between the language model and the physical-layer iteration. Simulations show that Sem-IR outperforms orthogonal time-division access (TDMA) and the same NOMA receiver without semantic feedback in block error rate (BLER), yielding an order-of-magnitude reduction over NOMA at 8 dB.

Clustering mixed tabular data requires a unified metric space to bridge the inherent heterogeneity between continuous numerical measurements and discrete categorical symbols. Traditionally, algorithms rely entirely on dataset-internal statistics to estimate categorical relationships, which confines the learned metric to empirical co-occurrences and ignores conceptually obvious yet statistically unobserved affinities. Although LLMs offer external world knowledge, applying their text-centric reasoning to highly abstract tabular concepts presents significant challenges. Bridging this modality gap to construct a semantically complete metric typically requires embedding LLMs into iterative metric learning loops to dynamically optimize cross-modality representations. This incurs intractable computational overhead, forcing a compromise between semantic enrichment and scalability. Therefore, we propose GRACE, an LLM-grounded framework for scalable mixed-data clustering. GRACE shifts semantic acquisition to the attribute-value level via a multi-perspective LLM querying strategy, mapping heterogeneous values into knowledge-informed descriptions. Crucially, this one-shot grounding extracts general-purpose semantic representations that embed heterogeneous attributes into a unified space, decoupling expensive LLM invocation from iterative optimization. Furthermore, GRACE cross-validates these external semantics against dataset-internal statistical evidence to ensure alignment with the dataset-specific cluster structure. Ultimately, GRACE matches the scalability of conventional statistics-driven baselines while achieving superior clustering accuracy and conceptual interpretability over 11 competing methods. The source code is available at https://github.com/develop-yang/GRACE-GRACE-A

In this paper, we propose a multistage decoding framework that leverages internal information produced by an underlying message-passing decoder. The proposed method targets the failure dynamics caused by both classical trapping sets and degenerate errors supported on symmetric stabilizers, which are among the primary limitations of iterative decoding for QLDPC codes. To identify unreliable variable nodes, we introduce a heuristic metric that combines several dynamical features of the decoder, including variable-node log likelihood reliabilities, hard-decision oscillations, the number of adjacent unsatisfied checks, and the soft information contributed by unsatisfied checks. Based on this ranking metric, the decoder performs guided rewinds by selectively forcing the initial log likelihood ratio values of the most suspicious variable nodes and restarting the message-passing decoder under the corresponding forced configuration. To manage the combinatorial growth of candidate configurations, the search is formulated within a beam- search framework with controlled beam width. In addition, we introduce a pruning metric based on the combination of the residual syndrome weight and a posteriori reliability of the decoder output, thereby retaining only the most promising search paths. Logical error rate results demonstrate that the proposed decoder significantly outperforms the normalized min- sum decoder and achieves competitive performance with belief propagation enhanced by order-10 ordered statistics decoding.

Many dynamical-state reconstruction problems seek to infer a high-dimensional spatial state from measurements at only a few locations. Because the governing dynamics couple evolution across space and time, temporal measurements can contain information about state components beyond the sensor locations. This work develops a theoretical framework comparing the information content of a temporal measurement history with that of a specified instantaneous spatial sensor array. The temporal history and spatial reference array are represented by prior-whitened information operators that account for prior variability, sensor geometry, and measurement noise while preserving the directional distribution of uncertainty reduction. A general space-time information-exchange theorem gives necessary and sufficient conditions for retaining a prescribed fraction of the spatial-reference information in every state direction. It answers three practical questions. First, how many temporal measurements and how much total information are required? Second, is the prescribed fraction attainable, or does a limiting information ceiling rule out every finite history? Third, when attainable, what history length is sufficient? Explicit history-length bounds are derived for contractive, spectrally separated unitary, and simultaneously diagonalizable dissipative linear dynamics, and for stochastic linear estimation under Gaussian process and measurement noise. For nonlinear systems, global delay-map injectivity is combined with a uniform local Fisher-information condition. Numerical experiments test directional exchange lengths, information ceilings, and finite-memory bounds, and assess nonlinear delay-map distinguishability and local Fisher-information predictions. The framework provides an explicit basis for determining when temporal measurements can reduce spatial sensor coverage for full-state reconstruction.

Building on advances in reconfigurable antenna techniques, movable antennas (MAs) can dynamically reshape antenna arrays and introduce additional spatial degrees of freedom (DoFs), thereby further improving communication performance. Despite these benefits, existing MA design algorithms often entail prohibitively high computational complexity from discrete positioning selection, which prevents practical implementations of MAs. In this paper, we investigate efficient solutions for the mutual information (MI) maximization problem of a multi-user multiple-input multiple-output (MU-MIMO) uplink communication system aided by discrete MAs. To this end, we first formulate the discrete MA positioning problem with the assumption of perfect channel state information (CSI). Then, we prove that the design problem falls into the category of monotone submodular maximization subject to a 2-system constraint. Accordingly, we propose a low-complexity distance-constrained submodular position search algorithm, which is theoretically shown to achieve at least 1/3 of the optimum. Furthermore, we extend our approach to scenarios with imperfect CSI, and show that the proposed submodular optimization-based design remains robust against channel estimation errors. Numerical results demonstrate that the proposed scheme can achieve at least 90% of the optimal solution's MI gain under both perfect and imperfect CSI assumptions. Remarkably, the algorithm achieves orders-of-magnitude complexity reduction (e.g., 34.4x faster than the branch-and-bound approach) while maintaining significant MI gains.

Let $C_m:=\mathrm{BCH}(3,m)$ be the binary primitive triple-error-correcting BCH code of length $2^m-1$. We determine its second generalized covering radius exactly: $R_2(C_m)=8$ for every $m\geq5$. Equivalently, every two-dimensional syndrome subspace is contained in the binary span of at most eight parity-check columns, and eight columns are necessary in the worst case. This matches the known lower bound.

This paper studies entanglement-assisted quantum locally recoverable codes (EA-qLRCs) built via a CSS-like stabilizer construction from pairs of classical locally recoverable codes (cLRCs), without requiring dual-containment. We define such codes through local recovery channels, give a sufficient stabilizer criterion for the construction, and derive Singleton-, Griesmer-, Plotkin-, and sphere-packing-like converse bounds on the parameters of the resulting pure CSS-like EA-qLRCs, along with a Cadambe–Mazumdar-like bound that, as in the classical case, lacks a closed form, plus a comparison of their relative tightness across finite-length and asymptotic regimes. We give necessary and sufficient conditions for a pure CSS-like EA-qLRC to attain the Singleton-like bound with equality; for the single-code case $\mathcal{C}_1=\mathcal{C}_2=\mathcal{C}$, this reduces to a simple condition on the hull dimension $s=\dim(\mathcal{C}\cap\mathcal{C}^\perp)$, which also fixes the entanglement count via $c=n-k-s$. We present CSS-like EA-qLRC constructions from classical LRC families—Tamo–Barg and cyclic codes—and characterize when these attain the Singleton-like bound, showing the cyclic families yield optimal codes while the Tamo–Barg construction, though valid, attains the bound only in the degenerate regime $k \le r$, where locality is vacuous. We complement these constructions with two Gilbert–Varshamov-like achievability bounds, via a classical parity-check augmentation and a sharper concatenated-code construction, and show both hold unconditionally for field size $q>3$ via a monomial-equivalence argument. Finally, we unify all bounds—converse and achievability alike—under a common maximally entangled regime, giving a single comparison of the achievable and forbidden rate–distance–locality region for CSS-like EA-qLRCs.

Classical likelihood-ratio tests and $\Delta$AIC exacerbate the statistical significance crisis by scaling with sample size, often flagging negligible improvements as highly significant. While causal estimands like the average treatment effect (ATE) quantify practical magnitude, their reliance on the expectation operator ties them to the data's original coordinate scale. Furthermore, existing pseudo-$R^2$ metrics are inadequate: variance-based measures ignore higher-order distributional changes, and current formulations lack invariance to monotone transformations. We resolve these limitations by introducing Entropic Variance (EV) as a rigorous, scale-independent generalization of error variance in ordinary least squares. We define the population EV-based parameter, $\rho^2_V$, which projects unbounded cross-entropy onto a standardized $[0,1]$ scale, and establish that the EV-based $F_\text{V}$ statistic asymptotically follows an $F$-distribution. Building on these distributional properties, we propose two estimators: the empirical population $R^2_{\text{SV}}$ and the out-of-sample predictive $R^2_{\text{SVP}}$. Both are derived by exponentiating per-observation cross-entropy and incorporate a degrees-of-freedom correction for training optimism. Leveraging the $F_\text{V}$-distribution, we derive refined $p$-values and confidence intervals for $\rho^2_V$ without requiring intractable Fisher information matrices. Simulation studies and a Parkinson's disease microbiome application demonstrate the superiority of variable selection via these EV-$R^2$ metrics. Notably, evaluating the $R^2_{\text{SVP}}$ of a LASSO path via data-splitting reduced false discovery rates from 80% to 6% in simulations while fully preserving signal recall.

We develop a one-shot (finite-blocklength) channel-coding framework based on the pairwise error probability (PEP) of a decoder with randomized tie-breaking. The tie-breaking rule yields a probability-integral-transform identity: the induced error spectrum describes both random-coding achievability and exact fixed-code converse statements, for an arbitrary decoding metric. We derive two variational identities for metric-weighted tail functionals of the PEP, one through the Neyman-Pearson $\beta$-functional and one through a reverse channel, valid for an arbitrary metric. Under matched maximum-likelihood decoding they specialize to representations of the spectrum itself, which is then jointly convex in the testing level and the input prior; combined with the reverse-channel representation, this gives a linear program for the prior-optimized minimax meta-converse – finite-dimensional in general and, for memoryless channels with fixed alphabets, of size polynomial in the blocklength after a type reduction. Prior optimization of the random-coding bound is formulated as a concave program over input distributions with an explicit gradient, solved by a direct first-order method. The framework recovers several classical one-shot bounds, including the random-coding union bound and minimax meta-converse of Polyanskiy-Poor-Verdu, the information-spectrum bounds of Han-Verdu, and the linear-programming converse of Matthews. Numerical examples on the AWGN and binary Z-channels illustrate the achievability-converse comparison and the effect of prior optimization.

This paper studies the Shannon capacity of lexicographic products of finite simple graphs, together with the Lovász theta function and the fractional Haemers number. The Shannon capacity is proved to be supermultiplicative under lexicographic products in either order, and these products are compared with the strong product. We explicitly construct three countably infinite families of lexicographic powers based on the Schläfli graph, the McLaughlin graph, and its second subconstituent; in each family, pairing each member with its complement yields strict supermultiplicativity and arbitrarily large multiplicative gaps. Bounds and exact-capacity criteria for lexicographic products are derived, and the resulting upper bounds are shown to be incomparable. The capacities of lexicographic products involving Kneser graphs, their complements, and $q$-analogues of Kneser graphs are determined. It is also shown that a lexicographic product with a complete outer factor preserves the Shannon capacity of an arbitrary inner factor. The capacities of iterated lexicographic powers are determined, including those of self-complementary graphs that are vertex-transitive or strongly regular. Elementary, self-contained proofs are also given for three known results: the multiplicativity of the Lovász theta function and the fractional Haemers number under lexicographic products, and the equality of the fractional and ordinary Lovász theta functions. Finally, an open problem concerning the Shannon capacities of lexicographic and strong products is posed.

This paper evaluates the Linked Autonomous Interplanetary Satellite Orbit Navigation (LiAISON) method in near two-body asteroid systems. While LiAISON's usefulness has been proven in multibody environments such as the cislunar region, the application of a two-body system with gravity perturbations has not been sufficiently demonstrated. Vesta and Ceres are good candidates to test LiAISON, because the asteroids have sufficiently strong perturbations to use LiAISON method. The study utilizes a Multi-Model Adaptive Estimation (MMAE) framework integrated with Unscented Kalman Filters (UKF) to estimate the spacecraft states and perturbations using only range and range rate measurements. Results from the Monte Carlo analysis indicate that the proposed framework makes accurate and consistent estimates of position, velocity, and the perturbations for cases of Ceres and Vesta.

The Karhunen–Loève transform (KLT) of the stationary AR(1) source is the classical optimality benchmark of transform coding. Its frequencies are roots of the transcendental equations of Ray and Driver, and the transform has therefore long been treated as an unstructured dense matrix. Rather than compute the exact KLT, the field turned to fast approximations of it, most famously the discrete cosine transform (DCT). This paper shows that the retreat was premature. The exact AR(1) KLT admits direct recursive factorizations, at every order $N\ge2$ and every correlation coefficient $\rho\in(0,1)$: at even orders, the transform reduces to a butterfly stage, two half-order copies of itself, and orthogonal corrections generated by rank-one boundary perturbations of the covariance; at odd orders, to two copies of the KLT of the one-sided prediction residual and an arrowhead stage absorbing the center sample. The residual KLT recurses through half order as well. All correction stages are Cauchy-structured, and applying them by the fast multipole method yields exact-KLT algorithms of $O(N\log N)$ complexity—the same order as the FFT and the fast sinusoidal transforms. The factorizations follow from the covariance matrix by the Sherman–Morrison identity and the classical secular eigenvalue updates, and degenerate, as $\rho\to1$, to the known parity splittings of the DCT-II. As secondary results, the factorization constants are shown to be algebraic functions of $\rho$—in radicals for all $N\le8$—and complete exact KLT modules for $N=2,\dots,8$ are given.

A cyclic $n$-root is an $n$-dimensional complex vector that solves a particular set of multivariate homogeneous polynomial equations. There is a one-to-one correspondence between unimodular cyclic $n$-roots and bi-unimodular vectors (CAZAC sequences) with leading entry one. It was conjectured by Björck and Saffari that the set of cyclic $n$-roots is finite if and only if $n$ is square free. It is known that such a set is infinite if $n$ is not square free, and finite if $n$ is prime. A critical reduction in Haagerup's proof for prime $n$ is to show that infinity of cyclic $n$-roots (for any $n$) implies existence of two vectors with disjoint supports in both the time domain and the frequency domain. In this paper we show that such pair of vectors always exist if $n$ is composite, indicating that the original reduction is not adequate for composite square free cases. A discussion on the existence of a single vector whose support is disjoint with its Fourier transform is also included.

Despite advances in artificial intelligence (AI) across multiple sectors, today's AI tools, including deep learning and generative AI, still fail when embedded into physical systems, such as robots and vehicles operating under real-world physical laws. This stems from their inability to maintain reliable world models for long-horizon planning under uncertainty and generalize to unseen scenarios. In this context, wireless networks, through pervasive sensing and communication, can orchestrate physical intelligence. However, current architectures optimize throughput, latency, and reliability and cannot support real-time physical AI coordination, requiring agents to maintain shared spatiotemporal context. To address these challenges, a network of holonic digital twins (HDT-Nets) framework is proposed to deliver real-time physical AI inference through holonic agents that actively reason about their environment rather than passively mirror physical assets. Each HDT is realized as a hierarchical structure spanning the physical agent and network edge, reasoning autonomously at the local level while cooperating with neighboring HDTs to form collectively intelligent units. In HDT-Net, causal Markov blankets spanning sensing, communication, and control determine which agents must coordinate and enable counterfactual reasoning over multi-domain interventions. Active inference within these boundaries unifies perception, action, and learning by minimizing expected free energy while deciding which beliefs to transmit based on their cognitive value to the receiver. Category theory ensures that transmitted beliefs preserve semantic structure across heterogeneous agents with incompatible representations. Finally, integrated information theory quantifies when collective intelligence exceeds independent operation and how network intelligence evolves through coordinated learning and information exchange.

Collective decisions in agent networks emerge from repeated local interactions, as agents update their opinions (colors) based on their neighbors, potentially leading to consensus. In sparse communication graphs, repeated interactions can amplify both the initial majority preference and the random fluctuations caused by individual agents. Understanding when the initial majority determines the final preference and how quickly an agreement emerges is, therefore, a basic question in collective decision-making. We study binary majority dynamics as an idealized model of this process. At each round, the interaction graph is independently resampled from the Erdős–Rényi model $\mathbb{G}(N,p)$, with $p=b\log N/N$ and fixed $b>1$. Each agent adopts the majority opinion among its neighbors, retaining its current opinion if there is a tie. For a fixed initial configuration, let $\Delta_0$ denote the initial difference between the number of blue agents and the number of red agents. We identify three regimes for the consensus time, determined by $|\Delta_0|$. First, when $\Delta_0\sqrt p=O(1)$, the probability of blue unanimity is $$ \Phi\left(\Delta_0\sqrt{2p/\pi}\right) + o(1), $$ where $\Phi$ is the cumulative distribution function of a standard Gaussian. Moreover, consensus is reached after $(1+o(1))\log N/\log\log N$ rounds with high probability. Second, in the intermediate regime $1\ll\Delta_0 \sqrt{p}\lesssim \sqrt{N}$, we establish explicit high probability upper and lower bounds on the time to blue-unanimity. Lastly, when the initial blue advantage is above an explicit constant multiple of $N/\sqrt{\log N}$, blue unanimity is guaranteed within two rounds with high probability. Our findings provide a complete description of how the initial advantage influences winner selection and the speed of consensus in this type of majority dynamics with sparse interactions.

Network coding allows intermediate nodes to encode received messages before transmission. The multiple-unicast conjecture asserts that coding has no throughput advantage over fractional routing for independent unicast sessions in any undirected network. Despite more than two decades of sustained study, this central open problem remains unresolved. The conjecture is deeply connected to computational complexity: a proof would yield long-sought lower bounds for fundamental problems. To study the conjecture, this paper develops a unified metric framework from the perspective that the basic objects behind the comparison between coding and routing are not cuts alone, but graph metrics. Using this framework, we prove the conjecture for three new classes of undirected networks: (a) networks with at most five terminal locations; (b) planar networks whose terminal locations lie on the boundaries of at most three designated faces, with each session's endpoints on one such face; and (c) networks with arbitrarily many nodes and terminal locations under a structural restriction on session endpoints. We give a new proof that the conjecture holds for networks with at most six coding nodes, without computer-aided search, and show that if the conjecture holds on $\Gamma_{3,3}$, then it holds whenever no three sessions have six distinct terminal locations.

The multiple-unicast conjecture asserts that network coding offers no throughput advantage over routing in undirected networks. Its validity is known to imply fundamental lower bounds in computational complexity. We propose a Session Interaction Framework that reduces the conjecture to a central equivalence: the conjecture holds universally if and only if every irreducible core is independent. This result transforms the global feasibility problem into a two-stage process. First, to make the reduction phase tractable, we provide simplified sufficient conditions for session dominance, offering geometric criteria to iteratively simplify complex session sets. Second, for the remaining "irreducible core," we propose a Session Decoupling Theorem, reducing the conjecture's validity for a session set to its independent subsets. Topologically, we prove that sessions separated by high-cost cuts or cut-vertices are guaranteed to be independent. By integrating these reduction and decomposition mechanisms, our framework offers a systematic methodology to verify the conjecture across general network topologies.

Among $2^{nR}$ length-$n$ random sequences, one sequence is an outlier whose index is random. Under hypothesis $\mathcal{H}_0$, the components of the outlier are independent and identically distributed (IID) according to $Q_0$, whereas under hypothesis $\mathcal{H}_1$ they are IID according to $Q_1$. The remaining $(2^{nR}-1)$ sequences are mutually independent, independent of the outlier, and IID according to $P$ under both hypotheses. Based on the observation of all $2^{nR}$ sequences, one wishes to decide between $\mathcal{H}_0$ and $\mathcal{H}_1$. Under the constraint that the decision error probability under $\mathcal{H}_0$ must be bounded away from $1$, we determine the fastest exponential decay rate of the decision error probability under $\mathcal{H}_1$.