Papers

Existing coded-computing designs do not explicitly exploit the intrinsic structure of the input data. In communication systems, statistical structure and redundancy are often removed through source coding (or compression) before channel coding is applied. This principle, however, does not transfer directly to coded computation. In many computational tasks, particularly in machine learning, the structure of the data is precisely what the computation seeks to exploit to infer outputs or learn meaningful patterns. Consequently, coded-computing schemes should preserve and leverage this structure in their code design, rather than ignoring or eliminating it through source coding. This observation motivates a different perspective on code construction. In many channel-coding schemes, such as Reed-Solomon codes, coded symbols are generated by evaluating a low-dimensional algebraic representation at selected points. In contrast, many high-dimensional datasets naturally concentrate near low-dimensional manifolds. In this paper, we exploit this intrinsic geometry by designing coded samples that follow the natural manifold of the data, rather than imposing an artificial low-dimensional structure unrelated to the data distribution. Inspired by graph-based manifold learning, we propose a manifold-aware encoding strategy for general coded computing (GCC). Experiments on neural network inference and high-dimensional polynomial evaluation demonstrate that the proposed strategy consistently and significantly reduces the mean squared recovery error under straggling compared with standard GCC.

BCH codes are among the most important classes of cyclic codes and have played a central role in coding theory and its applications. One of the fundamental problems in the study of BCH codes is to determine their exact minimum distances, which directly govern their error-correcting capability. Although the BCH bound provides a general lower bound, determining the exact minimum distance is often difficult, and many parameter families remain unresolved. In this paper, we investigate three conjectures and an open problem on binary BCH codes proposed by Chen, Xie, and Ding in \cite{Chen59}. We settle these conjectures on the exact minimum distances of three families of binary BCH codes by constructing codewords attaining the BCH bound. Beyond these conjectures, we further study the more difficult family codes and determine its minimum distance for some cases. We further study Open Problem 8.4: affirmative answers are obtained for the first two length families, while for the third family a sufficient condition is established and a counterexample shows that the unrestricted assertion does not hold in general.

In the torn paper channel, a codeword is cut at random locations, and the resulting error-free fragments are delivered to the decoder as an unordered multiset. Although the capacity of this channel can be achieved using random code, decoding such codes generally requires exponential time. The interleaved-pilot construction of Shomorony and Vahid embeds a De Bruijn sequence among the symbols of a shifted erasure code and aligns only fragments that are sufficiently long through global statistical uniqueness. A subsequent local-alignment scheme by Liu and Raviv employs run-length-limited constraints and exclusive all-zero markers to identify pilot positions from local structure, substantially reducing the minimum fragment length that can be aligned. We further improve the method of Liu and Raviv by replacing its fixed pilot sequence with a multilevel successive local alignment and pilot-recycling procedure. In Liu and Raviv, the pilot sequence is chosen once and must simultaneously balance the length of the pilot sequence against the ability to align short fragments. Our construction removes this limitation by reusing pilot sequences across successive decoding levels. A pilot sequence with an independent random linear code is first used to align and decode the longest fragments. The information recovered at this stage is then recycled as a larger pilot sequence for the next stage.This process is repeated over multiple levels, so that progressively shorter fragments are recovered while the length of the pilot sequence remains small. Our construction depends on choosing a series of random linear codes, and we show that for any $\varepsilon>0$, there exists a choice of such codes which attains rate of at least $\varepsilon$ below the capacity, with high probability as the block length goes to infinity. Therefore, our construction achieves the capacity of the torn-paper channel.

A pseudorandom code (PRC) is a keyed error-correcting code whose codewords are computationally indistinguishable from uniform strings. We study public-key PRCs over fixed alphabets against adversarial deletions, where the deletion channel may both depend on the public encoding key and the transmitted codeword. Let $\gamma_q^{\mathrm{LCS}}$ denote the asymptotic normalised longest-common-subsequence length of two independent uniform $q$-ary strings. We prove that for every fixed $q\ge2$ no multi-message public-key PRC with a single-output decoder is robust against all such $\delta$-deletion channels for any $\delta>1-\gamma_q^{\mathrm{LCS}}$. For $q=2$, the current rigorous bound $\gamma_2\ge0.792665992$ rules out every constant $\delta>0.207334008$. The proof uses pseudorandomness only to transfer an LCS event from uniform strings to independently sampled codewords and therefore also the resulting collision argument is information-theoretic and requires no secret key. We also extend the argument to list decoding: for every fixed constant $L$, provided the message space contains at least $L+1$ messages, no such PRC with output lists of size at most $L$ is robust for $\delta>1-\gamma_2^{(L+1)}$. Since $\gamma_2^{(m)}=1/2+\Theta(1/\sqrt m)$, these thresholds approach $1/2$. Our bounds are specific to public-key-dependent adversarial channels and do not apply to oblivious edit channels.

Let $X_1,X_2,\ldots$ be i.i.d. finitely supported random variables in a torsion-free abelian group, and write $S_k=X_1+\cdots+X_k$, and $H(S_k)$ is the Shannon entropy $S_k$, for all $k \ge 1$. We prove that, for every fixed $n\geq1$, \[ H(S_{n+1})-H(S_n) \geq \frac12\log\frac{n+1}{n} -o_{H(X_1)\to\infty}(1), \] uniformly over the ambient group and the input law. This proves a conjecture of Tao [29] in 2010.

Modern data systems are commonly studied through computational and information-theoretic methods, while their algebraic properties remain largely unexplored. This paper introduces a mathematical framework for modelling data systems using algebraic codes over finite groups, organized by an axiom signature: each algebraic axiom of the data domain is shown to be sufficient for a specific coding-theoretic capability, and each implication is given its precise reading (MDS parameters do not require a field; quotients require normality, not commutativity). Three types of results are presented. Proved: a Gilbert-Varshamov existence bound over arbitrary finite groups, commutative or not; explicit direct-sum codes over the elementary divisors of any finite abelian group; that transactions preserving a validity predicate form a submonoid of the full transformation monoid, which is not a group; that schema-preserving transformations form a finite group whose orbits are counted by Burnside's lemma; and that the constraint-preserving bijections of a data system form a finite group whose orbits partition the valid configurations. Demonstrated: an explicit code construction over $\mathbb{F}_5$, with known codes over $\mathbb{Z}_4$ and $(2^U,\triangle)$ cited to situate the framework. Proposed: an encoding efficiency measure Ef and an optimization functional $\Phi$ with tunable weights, both evaluated at declared coding instances. Weight-independent comparison holds under componentwise dominance; in general the weight simplex decomposes into polyhedral chambers on which the induced order is constant.

In this work we propose an approach to construct a fully asynchronous unsourced random access communications (URA) system. Besides the common URA features such as the absence of the user identification information in the packet and the decoding oriented to receiving the content of the messages, there is no limitation on the user transmission timing. The active users can transmit their packet any time on demand. The proposed system belongs to the class of preamble-payload URA formats. Contrary to the typical role of the preamble to serve as a temporary user identification, the preamble serves for the purpose of timing acquisition. The proposed system also employs a very small pool of preambles to resolve collisions of packets transmitted simultaneously.We provide a performance analysis and simulation results for the proposed system and demonstrate that for a wide range of signal-to-noise ratios (SNR)s the supported numbers of active users are close to the numbers of users supported by a slotted URA system counterpart.

This paper studies the problem of computing the stochastic probability (shortest code length) of the encoded vectors containing cluster structure using Normalized Maximum Likelihood (NML) model. This is of great theoretical and practical importance in data clustering based on Minimum Description Length (MDL) principle, such as for estimating the best number of clusters and best cluster structure for the data. Straightforward computation of the shortest code length of the vector containing cluster structure based on the NML model requires polynomial time with respect to the size of the vector and number of clusters. We show that this is a tractable problem by introducing a recursion formula for the efficient computation of normalizing constant from the NML model. The time complexity of the new formula is linear opposed to previous polynomial time with respect to the size of the vector and number of clusters.

We rethink the definition of privacy in multi-server, graph-replicated private information retrieval (PIR) systems, by introducing a novel setting where the user's privacy is governed by the servers' storage structure. In classical graph-replicated PIR, the user retrieves a single message stored at the servers, while hiding the message index from each server. In our proposed privacy setting, the user is concerned with hiding the message index from a particular server, only if that server stores the message being retrieved, and privacy is not imposed otherwise. We coin this relaxed privacy requirement as local user privacy and the resulting PIR problem as local PIR on the graph. Our focus is on two-replicated PIR systems, where every message is replicated twice and stored on two distinct servers. Specifically, we study local PIR systems where the storage is represented by simple graphs, i.e., every pair of vertices is associated with at most one edge, and by their multigraph extension, i.e., $r$ parallel edges replace every edge. For these settings, we establish bounds on the local PIR capacity, defined as the maximum number of message symbols retrieved, per downloaded symbol. The local privacy requirement yields significant capacity gain over the classical PIR capacity under the same storage structure. For instance, in settings where the graph is a disjoint union of multiple identical sub-graphs, the gain in the local PIR capacity over classical PIR capacity is multiplicative in the number of sub-graphs. Further, for connected graphs, we derive capacity lower bounds for edge-transitive and bipartite graphs, which are greater than the best-known PIR capacity bounds. From these and by establishing matching upper bounds, we exactly characterize the capacity for star graphs, cyclic graphs, and path graphs with odd number of vertices. We introduce two local PIR schemes for general graphs.

We study the optimal joint sampling and transmission policy for a transmitter communicating with two gossiping receivers that share information with each other, with the objective of tracking a source under the Version Age of Information (VAoI) metric. The transmitter can observe source-version changes, but it has to pay a sampling cost to get the current source information content. Similarly, it can communicate with a receiver by paying a transmission cost. Gossiping enables local information exchange and is able to reduce costly direct transmissions. With imperfect communication links, we formulate an infinite-horizon average-cost Markov Decision Process (MDP) to jointly minimize receiver VAoI, sampling cost, and transmission cost. Using Relative Value Iteration (RVI), we evaluate the optimal policy and establish several properties of its structure. We prove that sampling has a threshold structure in the transmitter VAoI. Among direct transmissions, it is optimal to serve the older receiver. We further characterize the transmit or idle decision through the receiver VAoI difference. Our analysis shows that link reliability and receiver VAoI imbalance have a significant effect on the optimal policy structure. Numerical results verify the structural properties and demonstrate the performance gains of the optimal policy over multiple baselines.

Watermarking has been proposed as a way to identify synthetic samples in estimation settings where no metadata is available to distinguish them from real samples, but its precise effects remain unexplored. In the absence of a distinguishing mechanism, it has been shown that adding synthetic samples significantly reduces the marginal efficacy of new real samples. In this work, we study the minimax loss of such recursive discrete distribution estimation in the presence of watermarks in contrast to the unassisted and oracle-assisted losses. When the fraction of real samples vanishes asymptotically, we provide a lower bound that shows that it is impossible to improve performance by adding watermarks unless the false negative rate of detection also vanishes. Additionally, we show that in most regimes, the worst-case losses of a sequence of simple deterministic estimators match the corresponding lower bounds up to constants. Finally, we propose masking, a randomization procedure that narrows the gap in the remaining regimes to a Jensen gap. We conjecture that a tighter lower bound argument can close this gap.

We design low-complexity, fault-tolerant decoders for quantum low-density parity-check (QLDPC) codes with the goal of reducing decoding latency. We target two major bottlenecks of decoding under the circuit-level noise model: (i) post-processing via order-statistics decoding (OSD), and (ii) the large number of auxiliary variable nodes commonly introduced to represent CNOT-induced correlations during syndrome extraction. Our key observation is that propagating CNOT faults (hook errors) create stabilizer-induced trapping sets (TSs) that are intrinsic to hypergraph-product (HGP) and lifted-product (LP) constructions. Therefore, instead of modeling each such fault with an explicit correlation node and relying on OSD to clean up the resulting failures, we design message-passing decoders that resolve the corresponding stabilizer-induced TSs directly. We obtain these decoders by deriving QLDPC decoders from decoders for the parent classical LDPC codes and using them collectively to correct broad families of stabilizer-induced TSs. For CNOT faults that manifest primarily as syndrome errors, we show that their effect is equivalent to a data error together with syndrome-bit measurement errors. Consequently, given repeated measurements and a decoding graph that already includes nodes representing syndrome-bit errors, no distinct variable node is needed for each CNOT fault. Using a phenomenological Tanner graph with nodes representing only data errors and syndrome-bit errors, simulations on the LP codes show a reduction in, or comparable, logical error rates relative to BP+OSD, at substantially lower decoding complexity.

Fix integers $e\ge2$ and $r\ge1$. In this paper we study the $r$-th generalized covering radius $\rho_r\left(BCH(e,m)\right)$ of the binary primitive $e$-error-correcting BCH code $BCH(e,m)$. By using an algebraic-geometric reformulation of the covering problem together with an explicit Lang-Weil estimate, we prove that \[\rho_r\bigl(\BCH(e,m)\bigr)\le(r+1)e-1\] for all sufficiently large $m$. For $e\ge7$, this improves a recent result of Belinsky–Zabokritskiy. Our proof gives a substantially simpler geometric approach to this upper bound. In particular it implies that \[\rho_2\bigl(BCH(e,m)\bigr)=3e-1\] for all sufficiently large $m$. Previously it was only known that \[\rho_2\bigl(\BCH(e,m)\bigr) \in \left\{3e-1,3e\right\}\] for all sufficiently large $m$.

We study coefficientwise permutation lifts of reduced polynomial representatives of almost perfect nonlinear (APN) permutations from finite fields to Galois rings. The standard permutation criterion shows that a permutation polynomial over $\mathbb F_{2^m}$ admits such a lift if and only if its formal derivative is nowhere zero on the field. In that case, every coefficientwise lift permutes $\operatorname{GR}(2^k,m)$ for every $k>1$. It is natural to conjecture that the reduced representative of an APN permutation admits no such lift. However, Zhang and Haobo supplied a counterexample to our earlier critical-point conjecture, which also refutes this lifting claim. Their degree-$24$ reduced APN permutation over $\mathbb F_{32}$ has a nowhere-zero formal derivative. We record this example and give an exact linear-image and trace-dual criterion for the absence of rational critical points in quadratic functions. We also obtain explicit non-APN bounds from Janwa-Wilson-Rodier surfaces. For every odd degree $d\ge5$ outside the Gold and Kasami-Welch exponent families, results of Hernando-McGuire and Aubry-McGuire-Rodier yield an absolutely irreducible component defined over the ground field. Applying the Cafure-Matera point estimate gives an explicit threshold above which no polynomial of degree $d$ over $\mathbb F_{2^m}$ is APN. This makes the known qualitative nonexistence result effective. An identity between difference tables extends the bound to polynomials $ax+g(x^2)+c$, where $a\ne0$ and $g$ has an odd degree in the stated range. Finally, every cubic permutation polynomial has a rational critical point and therefore admits no coefficientwise permutation lift.

In doubly selective channels, the joint delay and Doppler dispersion generally induces coupling among transmitted symbols, thereby increasing receiver equalization complexity. Nevertheless, by using appropriately designed waveforms, channels with certain delay-Doppler (DD) supports can be diagonalized for one-tap equalization. The whole picture of such DD supports and their corresponding waveforms is still unclear, except for several examples identified in literature. In this paper, under cyclic-prefix (CP)-based block transmission and assuming that the modulation waveforms form an orthonormal basis, we identify all such channel supports by an elementary expression, and derive the corresponding waveforms in closed form.

Terrestrial base stations (BSs) are typically configured with fixed downtilt to serve ground users, resulting in weak illumination of low-altitude airspace even under line-of-sight (LoS) propagation. In this paper, we establish a channel model that incorporates BS and intelligent reflecting surface (IRS) radiation patterns for three-dimensional (3D) low-altitude coverage while preserving the existing BS configuration. We formulate a budget-constrained IRS deployment problem that jointly determines candidate-site selection, IRS orientations, and phase shifts to maximize the worst-case signal-to-noise ratio (SNR) over the 3D low-altitude airspace. The selected sites and optimized IRS parameters remain fixed after deployment, yielding a quasi-static IRS configuration. We characterize the illumination geometry between the fixed-downtilt BS and rooftop candidates by deriving the nonnegative installation-height range satisfying the BS main-lobe condition. The separation between the mapped main-lobe height boundaries grows linearly with horizontal BS-to-site distance and decreases inversely with the number of BS antennas. We further derive an analytical lower bound on the regional worst-case normalized array gain achievable through IRS phase design over served directions with different direction spans. The resulting sufficient direction span decreases inversely with the square root of the number of IRS elements when the same worst-case normalized gain guarantee is maintained. We develop a mixed-integer alternating optimization (AO) algorithm to solve the resulting problem. Simulation results validate the analytical characterizations and show that the proposed scheme achieves higher worst-case SNR than benchmarks across different deployment budgets.

The read process of DNA-based data storage systems generates multiple noisy copies of the stored DNA sequences, affected by edit errors consisting of substitutions, deletions, and insertions. Motivated by the challenge of ensuring reliable data retrieval in the presence of edit errors, we present a concatenated coding scheme that accounts for practical design constraints in DNA storage. We introduce and apply the marker-delimited code (MDC) as the inner code, which enables fast and reliable computation of symbolwise a posteriori probabilities (APPs). We combine MDC with an outer LDPC code. The LDPC is decoded via belief propagation using the soft information generated by MDC. Our results show that, in comparison with prior work, this construction provides more efficient error correction over multi-read edit channels in the short-blocklength and high-rate regime.

Jaramillo-Velez recently introduced edge codes, a new class of toric evaluation codes constructed from the edges of a (hyper)graph $\mathcal{H}$. In the case that $\mathcal{H}$ is a tree, Jaramillo-Velez computed both the minimum distance and the weight distribution of the associated code. In this paper, we study edge codes associated to unicyclic graphs. Our most striking result is that computing the parameters of these codes is subtle in the case that the induced cycle has an even length because these values will depend on certain conditions regarding the length of the cycle and the size of the base field.

We construct smooth, strictly log-concave counterexamples to the Gaussian completely monotone conjecture in every dimension. In one dimension, they form an explicit family $f_m$ whose signed $m$th entropy derivative at time zero is negative for every sufficiently large $m$; the inequality persists for all sufficiently small positive times. Tensorization with a broad Gaussian factor gives the higher-dimensional examples. The argument is analytic and self-contained. It reduces the sign to a two-frequency entropy calculation on the circle and transfers the resulting asymptotic to the real line through an exact heat-flow formula for Gaussian-windowed Fourier modes. The proof was developed by GPT-5.6 Sol Pro under the authors' guidance.

The exact Shannon capacity is unknown for every odd cycle beyond the five-cycle $C_5$, making odd cycles a central open problem in zero-error information theory. Improving the known lower bounds requires constructing large independent sets in strong powers of these graphs. Recent AI-assisted work has produced a rapid sequence of improvements: building on the construction of Itty et al., Gao developed a recursive product construction for combining structured independent sets, and Buys, Polak, and Zuiddam (BPZ) subsequently strengthened this through a richer recursion framework. We continue this line of AI-assisted exploration and introduce a heterogeneous refinement of these constructions. The central observation is that the usefulness of an intermediate construction depends not only on the size of its current main independent set, but also on the auxiliary structure it carries into subsequent recursion. Consequently, different parts of that auxiliary structure need not use the same independent set, and different occurrences in a recursion need not use the same intermediate representation. We formalize this for Gao's binary product and derive explicit propagation rules showing how heterogeneous choices strengthen the resulting gadget while leaving its current code size unchanged, then extend the principle to the more general BPZ framework, tailoring constructions to the distinct roles they play within the recursion. Applying these refinements to the seven-cycle $C_7$, we obtain an independent set in $C_7^{\boxtimes 500}$ yielding $\Theta(C_7)\ge 3.25883262\ldots$, improving the best known lower bound. Beyond the numerical gain, the results illustrate a general principle for recursive zero-error constructions: intermediate structures with the same dimension and current code size can have different downstream value depending on where and how they are used in the recursion.

Noise modulation (NoiseMod) encodes information in the variance of a noise-like waveform. Its noncoherent structure enables low-complexity links, but co-channel users are challenging because interference affects the same variance statistic used for detection. We study fluid antenna multiple access (FAMA) for two-level NoiseMod with independent random bits from each interferer, producing two possible nonzero interference variances. We adopt exact finite-sample maximum-likelihood energy detection from prior NoiseMod/TNM work. We derive the exact single-port moment-generating function and moments of the random-bit aggregate interference, including heterogeneous user powers, and prove that the port minimizing conditional BEP is $\arg\max_k H_k/(\sigma_w^2+J_k)$, where $H_k$ is desired-link power and $J_k$ is instantaneous bit-state-aware interference variance. For independent fading ports, the common interferer-bit vector couples per-port decision metrics; conditioning on the number of high-state interferers yields an exact binomial-mixture order-statistic representation and avoids the generally invalid rule $F_{Z_{\max}}=F_Z^{N_p}$. For spatially correlated operation, an integer-$\mu$ $\kappa$-$\mu$ benchmark uses full pairwise Jakes correlation in scattered fields, with channel-averaged BEP evaluated by conditional Monte Carlo. A standard-error-aware load study shows increased admissible co-channel users as spatial degrees of freedom grow. Finally, a finite-sample variance-domain sensing diagnostic quantifies the oracle gap under frozen channels and interferer states, providing a lower-bound warning on fast-FAMA sensing difficulty rather than a deployable acquisition protocol.

Spectrum governance requires rules to be translated into machine-executable decisions while preserving incumbent protection, regulatory authority, and an auditable record of why a decision was made. We present SPECTRA-GOV, a Tri-Layer Adaptive Governance Architecture (TLAGA) connecting international treaty coordination, national adaptive licensing, and real-time enforcement. The work builds on the original reference implementation preserved at v0.1.0-paper and adds a post-audit evaluation layer rather than replacing it. V2 introduces controlled spectral-contention scenarios, corrected geodesic-distance calculation, per-operator selective authorization, paired baseline counterfactuals, policy perturbation, regulatory-change analysis, and causal provenance. Across 10,000 scenarios in each of seven contention classes, incumbent protection was 100.00% in the no-contention control, 99.92-99.87% in weak-to-dynamic classes, 99.47% under strong overlap, and 90.00% in the adversarial close-proximity class. In a paired S3 baseline experiment, selective authorization achieved 100% incumbent protection and 89.43% access opportunity, while the population-wide dynamic baseline achieved 99.91% protection and 99.53% access. Identical results for the dynamic-SAS and SPECTRA-GOV selective variants mean selective admission alone is not claimed as an exclusive algorithmic novelty. The contribution is the integration of policy representation, computational enforcement, auditability, provenance. Enforcement timing was measured in-process, with mean latency increasing from 0.028 ms for one operator to 1.682 ms for 500 operators; these are software benchmarks, not field measurements. The results establish a reproducible computational governance prototype and identify remaining questions before operational or regulatory claims can be made.

Anti-Latin squares were introduced in connection with non-linear secure network coding, and the extremal problem for large mutually orthogonal families is motivated by that setting. We study the maximum size $N_A(d)$ of a family of mutually orthogonal anti-Latin squares of order $d$. We prove that $N_L(d)+1\le N_A(d)\le N_L(d)+2$ for every $d\ge 3$, where $N_L(d)$ denotes the classical maximum size of a family of mutually orthogonal Latin squares of order $d$, and we show that in fact $N_A(3)=N_L(3)+1$ whereas $N_A(d)=N_L(d)+2$ for every $d\ge 4$. The upper bound is obtained by passing through balanced matrices, while the lower bound is given by a deterministic permutation argument. For all $d\ge 8$, and also for the exceptional order $d=6$, the upper bound is shown to be attainable by a general probabilistic construction. On the structural side, we show that a saturated family of size $d+1$ induces an affine plane of order $d$, and that the saturated case is characterized by the existence of an anti-coordinate grid decomposition; after transporting this condition to the fixed cell set $[d]^2$, it becomes a direction-completeness condition on the corresponding row-blocks and column-blocks. The remaining small orders are treated separately: $d=3$ is handled by direct analysis and classification of orthogonal triples, $d=4$ by an explicit saturated construction and an analysis of its finite-geometric structure, and $d=5$ and $d=7$ by explicit saturated examples arising from the random-grid framework. Thus $N_A(d)$ is determined in terms of $N_L(d)$ for every $d\ge3$, and its numerical value is obtained explicitly for every $3\le d\le9$.

The generalized covering radii have recently emerged as fundamental parameters of linear codes with applications to database linear querying. In this paper, we study the generalized covering radii $\rho_t(M(m,q))$ of Melas codes $M(m,q)$ over any finite field $\mathbb{F}_q$. We determine $\rho_2(M(m,q))$ for all $q$, and for a general $t \ge 3$, we prove that $\rho_t(M(m,q)) \in \left\{2t,2t+1\right\}$ for $q \in \{2,3\}$ and $\rho_t(M(m,q))=2t$ for $q \ge 4$ whenever $m$ is sufficiently large. These results extend recent work on the covering radius of Melas codes.

In this work, we propose an efficient decoding algorithm for codes arising from simplicial complexes, a family of binary linear codes for which no decoding method of this type was previously known. Although the algorithm does not always attain the maximum theoretical error-correcting capability, it provides an explicit bound that can be computed directly from the structure of the complex. Moreover, this bound is asymptotically optimal: the ratio between the guaranteed correcting capability and the theoretical maximum converges to $1$ as the code length increases, under natural assumptions on the dimension of the maximal faces. The correction capability is also presented in specific examples. Finally, we introduce specific families of simplicial complexes where the algorithm successfully reaches this theoretical bound.

Multi-access speculative inference (Multi-SPIN) extends SPIN to multi-device edge networks to accelerate cooperative token generation. It allows on-device small language models (SLMs) to autoregressively draft multiple tokens for individual generation tasks, while an edge-server large language model (LLM) verifies them in parallel. The major communication overhead arises when a drafted token is rejected by the server, in which case sampling the correction token requires access to both the SLM-output draft distribution and the LLM-output target distribution over the full token vocabulary. Existing designs typically perform correction at the server by uploading the draft distribution, but transmitting a vocabulary-wide distribution creates a critical uplink (UL) bottleneck. Alternatively, the correction can be performed at the device by downloading the target distribution, leveraging the high transmission rates available on the downlink (DL). Motivated by this insight, we introduce communication-mode selection as a new design dimension for Multi-SPIN. Specifically, each device can adaptively switch between the UL and DL modes to balance the UL bottleneck against the shared DL resource constraint, thereby relieving the overall communication burden. We formulate a sum-token-goodput maximization problem that jointly accounts for mode selection, draft-length control, and power allocation. For mode selection, we reveal a simple optimal structure that enables efficient search over the number of UL devices, with the corresponding transmit powers optimized accordingly. For draft-length control, we develop a greedy-search algorithm that adapts device-specific draft lengths to heterogeneous computation and communication capabilities. Experimental results on Qwen2.5 and DeepSeek-R1 model pairs demonstrate that the proposed framework significantly improves token goodput.

Function correcting codes (FCCs) are designed to protect a specified function evaluation of messages at a higher level than the level of protection for messages, against errors while reducing the redundancy required for reliable communication. FCCs have thus far been studied for channels matched to various distances, including the Hamming and Lee distances. Every function partitions the message space into preimage sets corresponding to its distinct function values. Existing Plotkin-type bounds on the optimal redundancy of FCCs under the studied distances, applicable to arbitrary functions on the message space, depend on the pairwise distances among all the message vectors. This makes these bounds difficult to compute. We derive a general Plotkin-type bound on the optimal redundancy of FCCs under Wyner Graham distances, which include the Hamming and Lee distances as special cases. Our bound depends only on the cardinalities of the preimage sets and the sum of pairwise distances only among vectors within each preimage set. This approach significantly reduces the computations required to evaluate the existing Plotkin-type bounds and yields simplified bounds that are easier to compute for specific functions. We obtain simplified Plotkin-type bound for linear functions under the Wyner-Graham distance. Furthermore, the existing simplified bounds for linear functions under the Hamming and Lee distances are recovered as special cases of the proposed bound. We also obtain simplified bounds for several important classes of functions, including the Hamming weight function, the Hamming weight distribution function, the monomial functions under the Hamming distance, and the modular sum function, the Lee weight function, and the Lee weight distribution function under the Lee distance.

We study the design of linear time-invariant (LTI) encoder-decoder pairs for transmitting the state of a discrete-time LTI vector source over power-constrained parallel Gaussian channels with feedback. Two types of power constraints are considered. Under individual subchannel power constraints, a necessary and sufficient condition for designing an encoder-decoder pair that achieves bounded estimation error covariance (EEC) is established via two coupled majorization inequalities involving the subchannel signal-to-noise ratios and the antistable poles of the source. Under total channel power constraint, we derive the minimum total power required for a feasible encoder-decoder design by exploiting partial-order progamming under majorization order. An analytical optimal power allocation is obtained for the case of equal noise variances, which admits a water-filling interpretation; for general noise case, a sequential water-filling algorithm is developed. Our results reveal that the difficulty of transmitting a discrete-time LTI source via LTI coding is governed not only by its topological entropy, but also by the evenness of the log-magnitudes of its antistable poles. The design methods for feasible encoder-decoder pairs are also provided.

We study generalized Hamming weights along the one-point code flag of an AJ-Gorenstein curve. We organize these weights in a graded array, the zero diagram, whose entries are generalized coweights: the largest numbers of evaluation points on which subcodes of prescribed dimensions vanish simultaneously. Twisted and Wei duality show that each row of the zero diagram determines both the generalized Hamming weights of a lower block of short codes and the missing weights of a reflected upper block of long codes in the GHW diagram of the complete flag. Our main quantitative result is a uniform coverage theorem. For an AJ-Gorenstein curve of genus $g$ and evaluation length $n>2g$, the proportion of generalized-weight positions determined exactly throughout the complete flag satisfies $\operatorname{Cov}_{\mathrm{full}} \ge \frac{n(n-1)+4g}{n(n+2g-1)}>\frac12$. Thus more than half of all generalized-weight positions in the complete flag are determined uniformly. For the smallest Suzuki curve, the general and Castle-specific mechanisms together determine $2{,}280$ of the $2{,}912$ positions, giving an exact coverage of $78.30\%$.

This article studies transmissive reconfigurable intelligent surface (RIS)-assisted architectures. It compares them with electronically steered phased arrays for the deployment of vehicular direct-to-satellite (D2S) communications in future satellite networks. Rather than treating RIS as a direct replacement for phased arrays, we clarify the operating regimes in which RIS can serve as a low-power wavefront-shaping aperture and those in which phased arrays remain preferable because of their high gain and mature beam-tracking capability. Moreover, phased arrays can support multi-beam operation, which is particularly beneficial for dual connectivity and seamless handover. We distinguish analog, digital, and hybrid phased arrays, discuss the relationship between transmissive RIS and reconfigurable transmitarrays, and highlight practical profile, tracking, and link-budget constraints for mobile terminals. The comparison shows that passive RIS offers attractive power efficiency and aperture scalability, active RIS can partially improve the link budget, and phased arrays remain preferable for high-throughput.