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math.IT: explore 333 source-linked works published from 2021 to 2026, with original documents and citations.

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Includes records with this source-supplied label or an explicit phrase match in their metadata. Matches indicate a mention, not proof that a paper uses a method or tests a material. Source versions are consolidated by DOI.

Sources: arxiv. Collection updated 2026-09-16. Counts describe this index, not the complete source archives.

arXiv2026

Rethinking quantum smooth entropies: Tight one-shot analysis of quantum privacy amplification

We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. A key role is played by the measured smooth Rényi relative entropy of order 2, which we show to admit an equivalent variational form: it can be understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on extractable randomness and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of privacy amplification, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured Rényi divergences, tightening the bounds of [Dupuis, arXiv:2105.05342] and recovering state-of-the-art asymptotic i.i.d. error exponents. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., arXiv:2202.11590] and proving its optimality for all hash functions.

quant-ph↗
arXiv2026

Shared Sky, Shared Spectrum: Coordinated Satellite-5G Networks for Low-Altitude Economy

Driven by both technological development and practical demands, the low-altitude economy relying on low-altitude aircrafts (LAAs) is booming. However, neither satellites nor terrestrial fifth-generation (5G) networks alone can effectively satisfy the communication requirements for ubiquitous lowaltitude coverage. While full integration of satellites and 5G networks offers theoretical benefits, the associated overhead and complexity pose significant challenges for rapid deployment. As a more economical and immediately viable alternative, this paper investigates partially-integrated networks where satellites and 5G systems operate with coarse synchronization yet achieve coordinated spectrum sharing, pooling their capabilities to jointly serve LAAs. Leveraging the inherent position-awareness of LAAs, we propose a framework for joint time-frequency spectrum sharing with an adaptive synchronization time scale, where only large-scale channel state information (CSI) is required. To avoid solving the NP-hard optimization problem directly, link-feature-aided clustering is employed following a divide-andconquer strategy. The proposed framework achieves substantial performance gains with low overhead and complexity, enabling swift advancement of low-altitude applications while paving the way for future integrated satellite-terrestrial network evolution.

cs.IT↗
arXiv2026

Explicit Rank Extractors and Subspace Designs via Function Fields, with Applications to Strong Blocking Sets

We give new explicit constructions of several fundamental objects in linear-algebraic pseudorandomness and combinatorics, including lossless rank extractors, weak subspace designs, and strong $s$-blocking sets over finite fields. Our focus is on the small-field regime, where the field size depends only on a secondary parameter (such as the rank or codimension) and is independent of the ambient dimension. This regime is central to several applications, yet remains poorly understood from the perspective of explicit constructions. In this setting, we obtain the first explicit constructions of lossless rank extractors and weak subspace designs for $r\ll k$, where $r$ denotes the rank (or codimension), over finite fields $\mathbb{F}_q$ with $q \ge \mathrm{poly}(r)$ and $q$ non-prime, with near-optimal parameters. For other finite fields, including prime fields and small fields, we obtain weaker but still improved bounds. As a consequence, we construct explicit strong $s$-blocking sets in $\mathrm{PG}(k-1,q)$ of size $O(s(k-s)q^s)$ for all sufficiently large non-prime fields $q \ge \mathrm{poly}(s)$, matching the best known non-explicit bounds up to constant factors. This significantly improves the previous best bound $2^{O(s^2 \log s)} q^s k$ of Bishnoi and Tomon (Combinatorica, 2026), which requires $q \ge 2^{Ω(s)}$. Our approach is primarily algebraic, combining techniques from function fields and polynomial identity testing. In addition, we develop a complementary Fourier-analytic framework based on $\varepsilon$-biased sets, which yields improved explicit constructions of strong $s$-blocking sets over small fields.

cs.IT↗
arXiv2026

Mixed-dimensional quantum MacWilliams identity: Bounds for codes and absolutely maximally entangled states in heterogeneous systems

As emerging quantum architectures evolve into heterogeneous networks combining different physical substrates, such as qubits for logic and higher-dimensional qudits for robust communication, the traditional scalar metrics of quantum error correction become insufficient. To address this, we introduce a mathematical framework based on dimension multisets to characterize quantum error-correcting codes (QECC) and absolutely maximally entangled (AME) states in mixed-dimensional Hilbert spaces. By replacing scalar weights with multisets, we accurately capture the exact physical composition of error supports across these diverse systems. Our central result is the mixed-dimensional quantum MacWilliams identity, which establishes the formal algebraic relationship between Shor-Laflamme enumerators and unitary weight enumerators. From this foundation, we deduce the mixed-dimensional shadow identity and derive rigorous, generalized constraints on code parameters, explicitly formulating the mixed-dimensional quantum Hamming, Singleton and Scott bounds, and developing a linear program to systematically evaluate code viability. For the Singleton bound, a tighter bound that has no homogeneous analogue is derived for pure mixed-dimensional codes. Finally, we deploy this enumerator machinery to thoroughly analyze AME states, utilizing shadow inequalities to constrain their existence and introducing a combinatorial grid method for the explicit construction of mixed-dimensional tripartite AME states.

quant-ph↗
arXiv2026

Construction of Quantum Rank-Metric Codes Using Hermitian Orthogonality

Stacked quantum memory is an architecture in which multiple layers of qubits are stacked. Quantum rank-metric codes are effective for error correction in stacked quantum memories. However, the previously proposed quantum Gabidulin codes based on the CSS construction had a problem: due to algebraic constraints, the applicable memory layouts were strictly limited to square shapes of odd length. In this paper, we first propose a framework for constructing quantum rank-metric codes from classical linear codes with symplectic self-orthogonality. Building upon this, we propose a new construction method for quantum Gabidulin codes by combining the Hermitian self-orthogonality of classical Gabidulin codes--utilizing the self-dual basis that exists when the extension degree of the finite field is even--with the quantum code construction method using Hermitian orthogonality by Matsumoto and Uyematsu. The proposed method succeeds in approximately doubling the ratio of the minimum rank distance to the number of physical qubits while maintaining the code rate. Furthermore, it eliminates the restriction of the conventional method that requires the number of cells and layers of the stacked memory to be odd, realizing the construction of quantum rank-metric codes applicable to memories with an even number of cells and layers. This construction improves the relative error correction capability of the stacked quantum memory architecture and increases the degree of freedom in design while preserving the code rate.

quant-ph↗
arXiv2026

Geometric mean and Lebesgue-type decomposition of completely positive maps

We introduce the geometric mean and the parallel sum of completely positive (CP) maps between von Neumann algebras, based on the Pusz--Woronowicz theory of positive sesquilinear forms. We provide a concrete characterization via a block matrix positivity condition and establish their fundamental properties, including the AM--GM--HM inequality with respect to the CP order. In finite-dimensional settings, our construction is compatible with the Choi--Jamiolkowski correspondence, under which the geometric mean of CP maps corresponds to the Kubo--Ando geometric mean of their Choi matrices. This yields a natural operator-theoretic framework for interpolating quantum channels. As an application, we obtain index-type inequalities for conditional expectations in subfactor theory. Finally, we establish a Lebesgue-type decomposition of CP maps via a parallel sum construction, thereby providing a unified framework that simultaneously generalizes Ando's decomposition of bounded positive operators and Kosaki's decomposition of normal positive functionals on von Neumann algebras.

math.OA↗
arXiv2026

Recovery Models for Linear Batch Codes

Various types of recovery algorithms for batch codes have been considered previously, including asynchronous recovery and recovery algorithms arising from Almost Affinely Disjoint (AAD) families. We initiate a systematic investigation of linear batch codes equipped with recovery algorithms. We introduce online and strongly online batch codes, investigate their relations with asynchronous batch codes, and introduce (m,L)-strongly online batch codes, which provide particularly simple recovery algorithms. We then study simple batch codes associated with arbitrary bipartite graphs, obtaining graph-theoretic conditions for such codes and generalizing several known results.

cs.IT↗
arXiv2026

Semantic Communication for Non-Terrestrial Networks: A Survey from Platform Constraints to System Design

Sixth-generation networks are expected to extend connectivity beyond terrestrial infrastructure through non-terrestrial networks (NTNs) comprising satellites, high-altitude platform stations, and unmanned aerial vehicles. However, these platforms operate in a regime that bit-fidelity-centric design handles poorly: high free-space path loss, massive round-trip delays, Doppler shifts of hundreds of kilohertz, limited visibility windows, and limited on-board computing capability compared with ground hardware. Semantic communication (SemCom), which transmits task-relevant meaning rather than exact bits, provides a promising way to address these constraints. This survey examines SemCom for NTNs from the perspective of how semantic mechanisms support different parts of the communication system. We first map five structural NTN constraints onto the semantic mechanisms that can address them, and we show that each platform imposes a distinct constraint vector that selects among those mechanisms. We then propose a five-plane taxonomy covering semantic representation and on-board encoding, channel-adaptive transmission, semantic networking, resource management, and distributed learning with knowledge-base maintenance, together with a cross-cutting trust plane, and we review the literature within it. Finally, we summarize current standardization efforts and available research resources, and identify open problems and future research directions for SemCom in NTNs.

cs.IT↗
arXiv2026

A Quantum Method of Types

The method of types is a fundamental tool in classical information theory, with applications ranging from composite hypothesis testing and universal source coding to the capacity of arbitrarily varying channels. In this work we introduce an empirical operator acting as a quantum analog of the empirical distribution. We show that this empirical operator satisfies combinatorial and large-deviation bounds, which in combination describe a quantum method of types. As an application, we use our method to prove a universal achievability result for composite quantum hypothesis testing.

cs.IT↗
arXiv2026

Hermitian Hulls of Cyclic Codes and Exact Dual Distances of Extended Linear Codes

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

cs.IT↗
arXiv2026

Quantum Bicycle LDPC Codes with High $kd^2/n$ from Divisor-Driven Search

Bicycle (two-block circulant) quantum low-density parity-check (LDPC) codes include some of the best known small quantum codes, yet their design has relied on group-algebra formulations in which the dimension and distance are accessible only through matrix computation. We show that in the cyclic case the construction collapses into the polynomial ring $\F_2[x]/(x^{l}-1)$: self-orthogonality is automatic, the quantum dimension is read off from a polynomial gcd, and the minimum distance is certified exactly through the Calderbank correspondence to additive codes over $\F_4$, turning code search into an algebraically pre-filtered enumeration that reaches parameter regimes poorly covered by existing tables. A computer search based on this framework recovers the short codes $[[42,12,4]]_2$ and $[[62,12,4]]_2$ and produces a family of codes with competitive figure of merit $kd^2/n$, including $[[66,20,7]]_2$ with $kd^2/n=14.85$, above the bivariate bicycle code $[[144,12,12]]_2$ ($kd^2/n=12$) at less than half the block length, together with $[[46,2,8]]_2$, $[[66,2,9]]_2$, $[[66,4,8]]_2$, $[[66,6,8]]_2$ and, at $n=90$, $[[90,16,6]]_2$, $[[90,18,6]]_2$, $[[90,20,6]]_2$. An exhaustive census at $n=48$ delineates the boundary of this picture: we exhibit a $[[48,10,6]]_2$ code from a minimal $48$-element group (the Aydin--Tamo--Barg realization uses $72$ elements), and prove that distance $5$ forces a stabilizer-rank loss, which excludes $[[48,10,5]]_2$ from the weight-$8$ symmetric coset family. The framework thus opens a systematic route to bicycle-type quantum LDPC codes beyond the reach of group-theoretic searches, and identifies exactly where genuinely coset-theoretic phenomena begin.

cs.IT↗
arXiv2026

Solutions to Three Conjectures and an Open Problem on Binary BCH Codes

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.

cs.IT↗
arXiv2026

Manifold-Aware General Coded Computing for Straggler-Resilient Distributed Computing

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.

cs.LG↗
arXiv2026

MaskCode: Mask Transformer for Feedback-Assisted Coding With Linear Block Codes

Feedback-based coding schemes have demonstrated substantial performance gains over today's open-loop coding schemes. Unfortunately, these gains are usually achieved in idealized settings with perfect feedback. Over the last few years, machine learning-based schemes have been shown to be promising solutions for implementing feedback-based codes, particularly when combined with short-block-length open-loop error correcting codes (ECCs) in a concatenated coding structure. However, existing ML-based feedback schemes remain agnostic to the outer code's structure, potentially misallocating feedback resources on error patterns already correctable by the outer ECC. To address this, we propose MaskCode, a Transformer-based inner feedback code for concatenated coding systems, which explicitly incorporates structural knowledge of the outer linear block code into the inner feedback encoder design via two synergistic mechanisms: 1) a soft syndrome-based input that informs the encoder about potential parity constraint violations, and 2) a code-aware attention mask derived from the Tanner graph. We further show that end-to-end training with a differentiable belief propagation (BP) decoder offers no additional gain, as MaskCode's structure-aware design already internalizes the structural knowledge of the outer code; in fact, backpropagation through the iterative BP decoder introduces gradient explosion, which degrades rather than improves performance. Extensive evaluations on BCH and LDPC outer codes demonstrate that MaskCode consistently outperforms all baselines, achieving up to 1.5 dB SNR gain.

cs.IT↗
arXiv2026

Coding for Multiple Reverse-Complement and Palindromic Duplications

Reverse-complement (RC) and palindromic (PAL) duplications copy a length-$k$ block, reverse the copy, and insert it immediately after the original block; an RC duplication also complements the copied symbols. We study $q$-ary codes correcting $t$ such operations performed sequentially, so a later operation may copy symbols created by an earlier one. For fixed $q\geq2$ and $k,t\geq1$, every length-$n$ code $C$ for either channel satisfies $n-\log_q|C|\geq t\log_q n-O_{q,k,t}(1)$; for fixed $q,k$ and $1\leq t=t(n)=o(n)$ the lower bound is $t\log_q(n/t)-O_{q,k}(t)$. For a single RC error over an even alphabet with a fixed-point-free complement, the previously known RC-specific construction applies at odd $k$ and does not cover even $k$. For every even $k$ and any involutive complement, we give a coordinate-wise bijection that turns each RC duplication into a PAL duplication. Applying this bijection to every codeword therefore converts any $t$-error-correcting RC code into a PAL code of the same size, and conversely; encoders and decoders transfer by adding linear-time coordinate passes. For every even $k$, we also determine the maximum number of distinct descendants produced by exactly two errors from one source word. Words alternating between any two distinct alphabet symbols attain this maximum for PAL, and the bijection gives the RC maximizers. For both PAL and RC at fixed even $k$, we prove the existence of two-error-correcting codes with redundancy $4\log_q n+O_{q,k}(1)$. In the binary two-error problem, the converse gives $2\log_2 n-O_k(1)$, leaving a factor-two gap in the best existence bounds.

cs.IT↗
arXiv2026

Algorithm-Hardware Co-Design of a Lightweight PCG Equalizer with a Fixed Step Size for Massive MIMO

Coarse quantization in massive multiple-input multiple-output (MIMO) systems reduces power but causes clipping distortions. The Bayesian Expectation-Maximization (BEM) algorithm can recover clipped signals, but its matrix inversion and dynamic step-size evaluation are hardware bottlenecks. We propose a hardware-friendly one-step correction that uses the initial Jacobi-preconditioned Conjugate Gradient (PCG) direction with a fixed relaxation parameter. The resulting symbol-level update has an ultra-lightweight $\mathcal{O}(U)$ feed-forward datapath and approaches high-resolution reference detectors in the evaluated massive-MIMO setting. Our finite-dimensional analysis establishes the exact one-step descent law, proves that Jacobi normalization cancels the raw multiplicative near-far scaling while confining the loaded-system dependence to bounded attenuation factors, and gives verifiable sufficient conditions for fixed-step descent in terms of normalized channel coherence. System-level results indicate projected power savings for energy-efficient massive MIMO uplinks.

cs.IT↗
arXiv2026

Exact Payload-Decoupling Conditions for Pilot-Only BEM Channel Estimation With Application to OTFS

In high-mobility doubly dispersive links, basis expansion models (BEMs) reduce channel dimensionality, yet unknown payload symbols generally contaminate conventional matched-pilot channel estimates. This paper establishes the exact conditions under which such estimates become payload-independent and derives a pilot, guard, and data-placement rule that guarantees these conditions. We prove a necessary-and-sufficient zero pilot--data interference (ZPDI) condition under which the matched-pilot least-squares (LS) solution coincides with the maximum-likelihood (ML) estimator for the reduced pilot statistic. When ZPDI holds, estimation requires a single precomputed projection. When it does not, the estimate contains a deterministic, channel-scaled payload bias that persists at high signal-to-noise ratio. A disjoint-support rule, independent of the selected basis, realizes ZPDI through pilot, guard, and data placement. We then specialize the framework to orthogonal time--frequency space (OTFS) and examine its structural and performance consequences. With the generalized complex-exponential BEM (GCE-BEM), the ZPDI estimator remains within about $2$~dB of the perfect channel state information benchmark in bit error rate at speeds up to 500~km/h.

cs.IT↗
arXiv2026

List Size of Random Linear Codes: Optimality and the Limits of Average-Radius List Decoding

Guruswami, Li, Mosheiff, Resch, Silas and Wootters conjectured that, for every fixed prime power $q$ and $p\in(0,1-1/q)$, random linear codes of rate $1-h_q(p)-\varepsilon$ have list size $\frac{h_q(p)}{\varepsilon}\cdot(1+o(1))$ as $\varepsilon\to0$, both for list decoding and average-radius list decoding. The conjecture was known for $q=2$ and remained open for larger alphabets. We prove the conjecture for list decoding for every prime power $q$, with the sharper bound $L=\frac{h_q(p)}{\varepsilon}+O_{p,q}(1).$ In contrast, we disprove the conjecture for average-radius list decoding. We show that the required list-size leading constant is at least $p\left(1+\log_q\frac{(q-1)(1-p)}{p}\right)$, which is strictly larger than $h_q(p)$ whenever $q^p(1-p)>1$. Moreover, for $p=1-λ/q$ with fixed $λ>1$, the required leading constant can be as large as $\left(\frac{\logλ}{eλ}+o_q(1)\right)\frac{q}{\log q}.$

cs.IT↗
Compare source metadata on this page
WorkPublishedSource identifierSource
Rethinking quantum smooth entropies: Tight one-shot analysis of quantum privacy amplification2026-09-012603.04493arxiv
Shared Sky, Shared Spectrum: Coordinated Satellite-5G Networks for Low-Altitude Economy2026-09-012603.14196arxiv
Explicit Rank Extractors and Subspace Designs via Function Fields, with Applications to Strong Blocking Sets2026-09-012604.13431arxiv
Mixed-dimensional quantum MacWilliams identity: Bounds for codes and absolutely maximally entangled states in heterogeneous systems2026-09-012604.25790arxiv
Construction of Quantum Rank-Metric Codes Using Hermitian Orthogonality2026-09-012605.02571arxiv
Geometric mean and Lebesgue-type decomposition of completely positive maps2026-09-012605.06019arxiv
Recovery Models for Linear Batch Codes2026-10-152605.09748arxiv
Semantic Communication for Non-Terrestrial Networks: A Survey from Platform Constraints to System Design2026-09-152606.05216arxiv
A Quantum Method of Types2026-09-012606.27442arxiv
Hermitian Hulls of Cyclic Codes and Exact Dual Distances of Extended Linear Codes2026-10-142607.12242arxiv
Quantum Bicycle LDPC Codes with High $kd^2/n$ from Divisor-Driven Search2026-09-012608.09115arxiv
Solutions to Three Conjectures and an Open Problem on Binary BCH Codes2026-09-172609.00532arxiv
Manifold-Aware General Coded Computing for Straggler-Resilient Distributed Computing2026-09-012609.00552arxiv
MaskCode: Mask Transformer for Feedback-Assisted Coding With Linear Block Codes2026-09-012609.00715arxiv
Coding for Multiple Reverse-Complement and Palindromic Duplications2026-09-292609.00779arxiv
Algorithm-Hardware Co-Design of a Lightweight PCG Equalizer with a Fixed Step Size for Massive MIMO2026-09-012609.00890arxiv
Exact Payload-Decoupling Conditions for Pilot-Only BEM Channel Estimation With Application to OTFS2026-09-012609.00937arxiv
List Size of Random Linear Codes: Optimality and the Limits of Average-Radius List Decoding2026-10-142609.01070arxiv

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