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arXiv2026

Capacity Achieving Torn Paper Codes

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.

cs.IT↗
arXiv2026

Synchronization Strings over the Optimal Alphabet

Synchronization strings provide deterministic position labels for recovering coordinates after insertions and deletions. Haeupler and Shahrasbi introduced these objects, and subsequent work proved that four symbols suffice for some fixed parameter epsilon < 1, whereas two symbols cannot support arbitrarily long synchronization strings. We resolve the remaining ternary case: every length admits a ternary 2001/2002-synchronization string. Thus three is the exact minimum constant alphabet size. A computer-assisted refinement based on a larger 54-uniform family yields ternary epsilon-synchronization strings for every epsilon > 215/216. The previous four-symbol construction uses a ternary square-free backbone to exclude short repetitions and a fourth symbol to carry long-range synchronization marks. Our main technical contribution is a local-entropy transfer theorem: every square-free block-local source with a positive interval conditional min-entropy rate supports synchronization strings with a fixed gap. We instantiate this theorem using occurrence-wise branching in a Brinkhuis family. Every outcome remains ternary and square-free, while every long interval retains linear conditional min-entropy after all choices outside it are exposed. A deletion-ball estimate converts this entropy into an exponentially small probability of a near-complete common subsequence between adjacent intervals, and an asymmetric Lovasz Local Lemma enforces all interval constraints simultaneously. The same framework also yields exponentially many valid words, synchronization circles, and synchronization within a class of extremal square-free words. Adding constraints on distant intervals gives a Las Vegas construction in expected O(n^2 log^3(n+2)) time.

cs.IT↗
arXiv2026

Minimum Schubert Codewords and Second-Minimum Grassmann Codewords

In this paper, we give a classification of the minimum weight codewords of Schubert codes $C_α(\ell, m)$ by settling the conjecture proposed by Ghorpade and Singh in 2018, for all values of $q$ and all $α$. We use this classification to prove that a codeword of the Grassmann code $C(\ell, m)$ has the second minimum weight if and only if it is indexed by an element of $\bigwedge^{m-\ell}V$ that can be written as the product of a decomposable $(m-\ell-2)$-vector and an alternating $2$-vector of rank $4$. Finally, we give an enumeration of the second minimum weight codewords of the Grassmann code.

cs.IT↗
arXiv2026

Equivalence of Fixed-Rank and Rank-One Even-Order Symmetric Tensor Factorization

In the recent work of Barbier, Ko, and the second present author on sublinear-rank symmetric matrix factorization [Math. Stat. Learn. 9 (2026), 1-68], a key result is that, in the Bayes-optimal setting, the large-size limit of the free entropy of the finite-rank spiked Wigner model is the same as in the rank-one case when the signal has centered i.i.d. entries. In this paper, we show that this rank-one equivalence result extends to the case of finite-rank, even-order, symmetric tensor factorization. Moreover, we give a natural reformulation of a hypothesis that was stated in the aforementioned work to be necessary for this result. As in the matrix case, we use information-theoretic identities and replica symmetry to reduce a known multi-dimensional variational formula for the limiting free entropy to its one-dimensional analog. The novelty stems from the fact that said formula involves a replica symmetric potential containing Hadamard (entrywise) powers, rather than squares, of the matrix-valued variational parameter, so the eigenvalue-based approach used in the matrix case must be adjusted.

cs.IT↗
arXiv2026

Spatial-Code-Domain Grouped Index Modulation: Fluid-Antenna-Assisted System Design and BER Performance Analysis

Fluid antenna systems (FASs) provide reconfigurable spatial resources within compact apertures. In this paper, we introduce code-domain grouped index modulation (CGIM) and its spatial-code-domain extension, termed SCGIM, for FA-assisted transceivers. CGIM partitions the available orthogonal spreading codes into multiple subsets and jointly maps information onto their in-phase and quadrature indices and constellation symbols. In an Rx-FAS-assisted single-input multiple-output (SIMO) link, group-wise despreading separates the orthogonal code groups for parallel detection, while receive-port selection provides spatial diversity. SCGIM further associates interleaved Tx-FA port subsets with the code subsets, with the Tx-FAS conveying spatial-index information and the Rx-FAS providing selection diversity in a multiple-input multiple-output (MIMO) link. For SCGIM, we develop maximum-likelihood (ML), staged greedy (GD), and cross-domain index message-passing (CD-IMPD) detectors. CD-IMPD exchanges soft information over a cycle-free factor graph to account for the coupling between the spatial and code indices, requiring only one inward and one outward message pass. For CGIM, the BER is derived from the joint decision regions of the despread-domain observations and averaged over the Rx-FAS selected-gain distribution under Rayleigh, Nakagami-m, and additive white Gaussian noise channels. For SCGIM, an average-BER approximation is derived from a full-pair union bound using the selected-gain density ratio and exponentially tilted quadratic-form Laplace transforms. Simulation results validate the BER analysis and show that the proposed schemes achieve lower BER and higher throughput than the considered IM schemes, while CD-IMPD achieves near-ML BER performance with lower detection complexity.

cs.IT↗
arXiv2026

Conditional Fisher-Information Central Limit Theorems under Log-Concavity with Information-Theoretic Consequences

We establish conditional central limit theorems in Fisher information under log-concavity in every fixed dimension. For conditionally centered normalized sums, after whitening by the averaged conditional covariance, the averaged conditional Fisher information converges to the dimension if and only if it is finite at one convolution level. The scalar criterion follows as the one-dimensional case; we also provide an independent scalar proof based on a second-order continuity theorem for Fisher production on Gaussian-smoothed, tail-controlled classes. For the original sums, the averaged Fisher information matrix converges in operator norm to the inverse averaged conditional covariance. Consequently, the conditional relative Fisher information with respect to the limiting Gaussian law vanishes, and the Gaussian logarithmic Sobolev inequality yields convergence in conditional relative entropy and conditional entropy. We give two operational consequences. For any fixed finite-constellation low-power input, the first-order conditional mutual-information slope converges to the Gaussian-noise benchmark. For Gaussian signaling at any fixed signal covariance, the mutual-information gap from that benchmark is bounded by the conditional relative Fisher deficit and hence vanishes asymptotically.

cs.IT↗
arXiv2026

Cyclic Codes of Length 7_p^s over F_p^m + uF_p^m : Characterization, Duals, and Applications to Quantum and LCD Codes

Let $R_2 = \mathbb{F}_{p^m} + u\mathbb{F}_{p^m}$ ($u^2 = 0$), where $p$ is an odd prime and $m, s \in \mathbb{N}$. For $p \equiv 3, 5 \pmod 7$ with $\gcd(m, 6) = 1$, the cyclotomic polynomial $Φ_7(x)$ is irreducible over $\mathbb{F}_{p^m}$. This yields a direct sum decomposition $C = C_1 \oplus C_2$ for any cyclic code $C$ of length $7p^s$ over $R_2$, where $C_1$ has length $p^s$ and $C_2$ is a $7$-cyclotomic code of length $6p^s$. We classify $7$-cyclotomic codes into four disjoint generator-based types and calculate exact cardinalities using residue and torsion subcodes. Furthermore, explicit generators for the Euclidean dual codes $C^\perp$ are determined. As operational applications of these classified codes, we construct new families of quantum stabilizer codes via the CSS framework and establish parameter criteria for linear codes with complementary duals (LCD codes).

cs.IT↗
arXiv2026

A Fundamental Limit in Decentralized Decision-Making

In decentralized decision-making, several agents connected according to a network graph aim at solving a classification problem by collecting streaming observations. Due to decentralization, they run an iterative algorithm where, at each iteration, they can only exchange information locally with their neighbors. While decentralized estimation solutions have been shown to match the performance of optimal centralized systems, we show here that surprisingly this conclusion does not hold for decentralized decision-making. Specifically, we prove that the error probability for the best decentralized decision strategy exhibits an irreducible loss with respect to the optimal centralized classifier. This result establishes a fundamental limit for the performance of any decentralized decision strategy. We obtain an analytical relation showing that this limit is related to the interplay between decentralization and classification. The first aspect appears through the distances between the nodes in the graph, while the second aspect plays through the moment generating functions of the likelihood ratios that describe the decision problem. By applying the derived closed-form relation to different network topologies and inference problems, we observe some interesting and perhaps unexpected behavior emerging. In particular, we characterize the scaling law (with the network size) for the loss over popular network topologies, showing that the error probabilities might differ by orders of magnitude; and we examine how performance is affected by the relative distance between informative and uninformative agents over the graph.

cs.IT↗
arXiv2026

Resolving the Discontinuity of Continuous-Time AFDM Waveforms

Continuous-time affine frequency division multiplexing (AFDM) waveforms, constructed via frequency wrapping and phase correction, are known to be sample-wise equivalent to the widely adopted discrete AFDM framework. In this paper, we uncover a fundamental and previously overlooked flaw in this construction: its complex envelope is inherently discontinuous for generic chirp parameters. We show that these discontinuities are the direct cause of the high out-of-band emission (OOBE). To resolve this issue, we propose a fundamentally different continuous-time waveform, termed stepped frequency division multiplexing (SFDM). Unlike conventional approaches that allow continuous frequency variation, SFDM freezes the instantaneous frequency at the midpoint of the underlying chirp trajectory within each Nyquist sampling interval. This design yields a complex envelope that is strictly continuous over the entire symbol duration while preserving exact sample-wise equivalence with discrete AFDM. A unified spectral analysis reveals that the superior OOBE performance of SFDM stems from the absence of internal jump discontinuities, which otherwise dominate the far-out spectral roll-off. Numerical results confirm that SFDM consistently achieves significantly lower OOBE across a wide range of chirp rates.

eess.SP↗
arXiv2026

Resource-Constrained Semantic-Aware Remote Estimation with Overlapping Sensor Coverage

We study semantic-aware remote estimation of multiple finite-state Markov sources observed by K sensors with overlapping coverage. The sensors share a time-division multiple-access uplink and differ in transmission reliability, delivery delay, and transmission budget. In each slot, the scheduler jointly selects a source and one of its monitoring sensors, or remains idle, to minimize the long-run average cost of actuation error subject to global and per-sensor transmission-frequency constraints. We formulate this problem as a finite average-cost constrained Markov decision process. We show that the transmission resource functions have rank at most K, although the global constraint may still restrict the feasible region. Consequently, the Lagrangian depends only on K effective transmission costs, and an optimal constrained solution can be represented using at most K+1 deterministic policy-recurrent-class components. We further characterize the piecewise-affine concave Lagrangian value function and derive projected dual subgradient ascent over an explicitly bounded multiplier set. Numerical results illustrate the value-function structure, the need for policy randomization in a representative instance, and the interaction between global and per-sensor transmission budgets.

cs.IT↗
arXiv2026

The Neighbor Graph of Linear Complementary Dual (LCD) Codes

Linear complementary dual (LCD) codes form an important class of linear codes with applications in cryptography, classical error correction, and quantum coding theory. In this paper, we study the neighbor relation on LCD codes over finite fields and the graph induced by this relation, where two codes are adjacent whenever they intersect in codimension one. We determine the number of neighbors of an LCD code that are also LCD, and we use this result to analyze the structure of the corresponding neighbor graph. In particular, we prove its regularity over arbitrary finite fields and establish further regularity properties for its main structural subgraphs in the binary and odd-characteristic cases. These results provide a graph-theoretic framework for the study of LCD codes and reveal a strong combinatorial regularity in their neighborhood structure.

math.CO↗
arXiv2026

A Note on Binary Quadratic Systems and their relation to complexity theory

Deciding whether a system of multivariate quadratic equations over $\mathbb F_2$ has a solution is a classical NP-complete problem, and remains so for square systems, with as many equations as variables. The hardness of this problem is one of the cornerstones of nowadays post-quantum cryptography. Let $\MQ_0(n)$ and $\MQ_1(n)$ denote the sets of square quadratic systems in $n$ variables having respectively no solutions and exactly one solution. $\cup_{n\geq 2} \MQ_0(n)$ is a coNP-complete language, while $\cup_{n\geq 2} \MQ_1(n)$ lies in DP. It is known that $\lim_{n\to \infty} |\MQ_1(n)|/|\MQ_0(n)|=1$. Here we prove the explicit finite-$n$ bounds \[ |\MQ_0(n)|<|\MQ_1(n)| \le \left(1+\frac{1}{2^n-1}\right)|\MQ_0(n)|, \] More generally, let $Q_d$ be the space of polynomial functions $(\FF_2)^n\to\mathbb F_2$ of degree at most $d$, and let $α_k$ count square systems in $(Q_d)^n$ having exactly $k$ solutions. Then \[ α_0<α_1 \le \left(1+\frac{1}{2^n-1}\right)α_0\,, \qquad 2\le d\le n \,. \] The proof combines matroid and coding-theoretic methods. We interpret $(\FF_2)^n$ as the ground set of the evaluation matroid of $Q_d$, express $α_0$ and $α_1$ through characteristic polynomials, and use a Whitney-type sign-reversing involution to show that the only terms that can push $α_1-α_0$ below $α_1/2^n$ come from the elements of a matroid port. These are identified with minimal-support words of the Reed--Muller code $\RM(n-d-1,n)=\RM(d,n)^\perp$; the required estimate then follows from the MacWilliams identity, the minimum-distance bound $2^{d+1}$, and the even-weight structure of the code.

cs.IT↗
arXiv2026

A Novel Steganography Scheme Using Quantum Hilbert Transform

The main goal of steganography is to transmit hidden messages in legitimate-looking communication messages. Phase-domain information hiding, however, has not been fully explored for quantum systems. This work introduces a finite-dimensional Quantum Hilbert Transform (QHT) as a unitary phase operator based on the Quantum Fourier Transform. Using this construction, we develop a QHT-based quantum steganography scheme that embeds classical bits as weak signed phase perturbations of quantum cover states. Bob recovers the hidden message through binary state discrimination, block aggregation, and classical error-correcting decoding.

quant-ph↗
arXiv2026

Energy Estimation of the Hamming Slice and its Applications

Let $R=\mathbb{Z}/(2^n-1)\mathbb{Z}$, where $n\geq 3$, and let $S_w\subseteq R$ be the residues whose canonical $n$-digit binary expansion has Hamming weight $w$. We obtain, in particular, an asymptotic formula for the additive energy of $S_w$ \[ E(S_w)=\frac{\left|S_w\right|^4}{|R|}+ \mathcal{O}\left(|R|^3 n^{-3} \right), \] which holds uniformly in $w$. The error term is optimal in order, with a matching lower bound for $w=\lfloor n/2+\sqrt{n} \rfloor$. It follows that triple sums of arbitrary unit dilates have asymptotically uniform representation counts when $\prod_{j=1}^{3} \left|S_{w_j} \right| /\left(|R| n^{-3/5}\right)^3\to\infty$, and that double sums have asymptotically full support when $\left|S_{w_1}\right| \left|S_{w_2} \right|/\left(|R| n^{-3/4}\right)^2\to\infty$. In the proof, modular collisions are represented using a cyclic binary carry automaton; this appears to be a novel approach in this area of problems.

math.CO↗
arXiv2026

Rotatable Antenna Aided Mixed Near-Field and Far-Field Communications in the Upper Mid-Band: Interference Analysis and Joint Optimization

In this paper, we propose to leverage rotatable antennas (RAs) for improving the communication performance in mixed near-field and far-field communication systems by exploiting a new spatial degree-of-freedom (DoF) offered by antenna rotation to mitigate complex near-field interference and mixed-field interference. Specifically, we investigate a modular RA-enabled mixed-field downlink communication system, where a base station (BS) consisting of multiple RA subarrays communicates with multiple near-field users in the presence of several legacy far-field users. We formulate an optimization problem to maximize the sum-rate of the near-field users by jointly optimizing the power allocation and rotation angles of all subarrays at the BS. To gain useful insights into the effect of RAs on mixed-field communications, we first analyze a special case where all subarrays share the same rotation angle and obtain closed-form expressions for the rotation-aware normalized near-field interference and the rotation-aware normalized mixed-field interference using the Fresnel integrals. We then analytically reveal that array rotation effectively suppresses both interference types, thereby significantly enhancing mixed-field communication performance. For the general case involving subarray-wise rotation, we propose an efficient double-layer algorithm to obtain a high-quality solution, where the inner layer optimizes power allocation using the successive convex approximation (SCA) technique, while the outer layer determines the rotation angles of all subarrays via particle swarm optimization (PSO). Finally, numerical results highlight the significant performance gains achieved by RAs over conventional fixed-antenna systems and demonstrate the effectiveness of our developed joint design compared to benchmark schemes.

eess.SP↗
arXiv2026

Dualities of dihedral and generalised quaternion codes and applications to quantum codes

Let $\mathbb{F}_q$ be a finite field of $q$ elements, for some prime power $q$, and let $G$ be a finite group. A (left) group code, or simply a $G$-code, is a (left) ideal of the group algebra $\mathbb{F}_q[G]$. In this paper, we provide a complete group-algebraic description for the Hermitian dual code of any $D_n$-code over $\mathbb{F}_{q^2}$, where $D_n$ is a dihedral group of order $2n$ with $n$ not divisible by $\operatorname{char}(\mathbb{F}_{q^2})$, through a suitable Wedderburn-Artin decomposition of the group algebra $\mathbb{F}_{q^2}[D_n]$, and we determine all distinct Hermitian self-orthogonal $D_n$-codes over $\mathbb{F}_{q^2}$. We also present a thorough representation of the Euclidean dual code of any $Q_n$-code over $\mathbb{F}_q$, where $Q_n$ is a generalised quaternion group of order $4n$ not divisible by $\operatorname{char}(\mathbb{F}_q)$, via the Wedderburn-Artin decomposition of the group algebra $\mathbb{F}_q[Q_n]$. In particular, since the semisimple group algebras $\mathbb{F}_{q^2}[Q_n]$ and $\mathbb{F}_{q^2}[D_{2n}]$ are isomorphic, then the Hermitian dual code of any $Q_n$-code has also been fully described. As an application of the Hermitian dualities computed, we give a systematic construction, via the structure of the group algebra, to obtain quantum error-correcting codes, and in fact, with this methodical approach, we recover some already known quantum codes that achieve the best known minimum distance for their length and dimension.

cs.IT↗
arXiv2026

Fault-tolerant quantum processing of physical experiments

Quantum computers may reveal features of Nature inaccessible to conventional experiments, but manipulating raw quantum data introduces noise that degrades inference even when the processor is fault-tolerant, creating a data-input bottleneck for robust quantum learning. Here we show that quantum fault tolerance can substantially improve the sample complexity of learning from noisy experiments. We encode unknown quantum states from physical experiments into protected quantum memory, enabling fault-tolerant implementations of quantum learning algorithms otherwise degraded by errors. Using this quantum uploading procedure, we prove that noisy randomized measurement and multi-copy learning tasks can be performed exponentially faster than by any adaptive strategy that does not immediately encode physical states into error-corrected memory. These separations are not simply due to a reduced effective noise rate: they hold even when uploading is substantially noisier than the bare experimental interface, rigorously establishing immediate encoding as the optimal approach to noise-robust learning. We numerically illustrate the speedups in astronomical imaging, where quantum processing of uploaded photons locates an exoplanet obscured by a bright star using orders of magnitude fewer shots than unencoded baselines. Our results establish a robust interface between quantum computers and natural systems, enabling powerful and practical quantum-enhanced experiments.

quant-ph↗