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cs.IT

cs.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

Marker-Delimited Codes for Short-Blocklength, High-Rate Coding over Multi-Read Edit Channels

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.

cs.IT↗

arXiv2026

Intelligent Reflecting Surface Deployment for Low-Altitude Coverage: Illumination Geometry, Directional Characteristics, and Optimization

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.

cs.IT↗

arXiv2026

Reduced polynomial lifts of APN permutations over Galois rings and effective non-APN bounds

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.

math.NT↗

arXiv2026

Asymptotic Bounds on Generalized Covering Radii of Binary Primitive BCH Codes

Fix integers $e\ge2$ and $r\ge1$. In this paper we study the $r$-th generalized covering radius $ρ_r\left(BCH(e,m)\right)$ 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 \[ρ_r\bigl(\BCH(e,m)\bigr)\le(r+1)e-1\] 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 \[ρ_2\bigl(BCH(e,m)\bigr)=3e-1\] for all sufficiently large $m$. Previously it was only known that \[ρ_2\bigl(\BCH(e,m)\bigr) \in \left\{3e-1,3e\right\}\] for all sufficiently large $m$.

cs.IT↗

arXiv2026

Fast Fault-Tolerant Decoders for Hypergraph Product and Lifted-Product Codes

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 \emph{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 (\emph{hook errors}) create \emph{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 \emph{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 \emph{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 \emph{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.

cs.IT↗

arXiv2026

Minimax bounds for watermarked and masked recursive discrete distribution estimation

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.

cs.IT↗

arXiv2026

Semantic Freshness Optimal Sampling and Transmission for Gossiping Receivers

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.

cs.IT↗

arXiv2026

Local Private Information Retrieval for Graph-Based Replicated Systems

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.

cs.IT↗

arXiv2026

Stochastic complexity of vectors containing cluster structure

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.

cs.LG↗

arXiv2026

An Approach to Asynchronous Unsourced Random Access

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.

cs.IT↗

arXiv2026

An Algebraic Framework for Data Systems: Classification and Optimization of Algebraic Structures for Data Organization and Coding

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 \emph{axiom signature}: each algebraic axiom of the data domain is shown to be \emph{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 $Φ$ 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.

math.RA↗

arXiv2026

Shannon's problem on the monotonicity of entropy and a Conjecture of Tao

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.

math.PR↗

arXiv2026

A Public-Key-Dependent Adversarial-Deletion Ceiling for Fixed-Alphabet Multi-Bit Pseudorandom Codes

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 $γ_q^{\mathrm{LCS}}$ 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 $δ$-deletion channels for any $δ>1-γ_q^{\mathrm{LCS}}$. For $q=2$, the current rigorous bound $γ_2\ge0.792665992$ rules out every constant $δ>0.207334008$. 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 $δ>1-γ_2^{(L+1)}$. Since $γ_2^{(m)}=1/2+Θ(1/\sqrt m)$, these thresholds approach $1/2$. Our bounds are specific to public-key-dependent adversarial channels and do not apply to oblivious edit channels.

cs.CR↗

arXiv2026

Learning to Transmit Over Unknown Erasure Channels with Empirical Erasure Rate Feedback

We address the problem of reliable data transmission within a finite time horizon $T$ over a binary erasure channel with unknown erasure probability. We consider a feedback model wherein the transmitter can query the receiver infrequently and obtain the empirical erasure rate experienced by the latter. We aim to minimize a regret quantity, i.e. how much worse a strategy performs compared to an oracle who knows the probability of erasure, while operating at the same block error rate. A learning vs. exploitation dilemma manifests in this scenario -- specifically, we need to balance between (i) learning the erasure probability with reasonable accuracy and (ii) utilizing the channel to transmit as many information bits as possible. We propose two strategies: (i) a two-phase approach using rate estimation followed by transmission that achieves an $O({T}^{\frac 23})$ regret using only one query, and (ii) a windowing strategy using geometrically-increasing window sizes that achieves an $O({\sqrt{T}})$ regret using $O(\log(T))$ queries.

cs.IT↗

arXiv2026

Improving the decoding performance of CA-polar codes

We investigate the use of modern code-agnostic decoders to convert CA-SCL from an incomplete decoder to a complete one. When CA-SCL fails to identify a codeword that passes the CRC check, we apply a code-agnostic decoder that identifies a codeword that satisfies the CRC. We establish that this approach gives gains of up to 0.2 dB in block error rate for CA-polar codes from the 5G New Radio standard. If, instead, the message had been encoded in a systematic CA-polar code, the gain improves to more than 1.5 dB. Leveraging recent developments in blockwise soft output, we additionally establish that it is possible to control the undetected error rate even when using the CRC for error correction.

cs.IT↗

arXiv2026

Term Coding: An Entropic Framework for Extremal Combinatorics and the Guessing--Number Sandwich Theorem

Classical existence problems in extremal combinatorics ask whether finite operations can satisfy prescribed identities universally. Term Coding replaces this yes-or-no question by a graded one: for a finite system $Γ$, the maximum code size $S_n(Γ)$ is the largest number of satisfying assignments attainable on an $n$-element alphabet. We prove that normalisation and diversification associate $Γ$ with a labelled guessing game of guessing number $α$ and give finite-alphabet sandwich bounds. Consequently, $\log_n S_n(Γ)=α+o(1)$. Entropy and polymatroid inequalities provide systematic upper bounds. Examples include a five-cycle with exponent $5/2$, self-orthogonal Latin squares, and presentation-dependent exponents for universally equivalent identity systems. All theorems, lemmas and propositions in this paper have been machine-checked in the Lean 4 proof assistant; the development is available at https://github.com/SR123/term-coding-lean.

cs.IT↗

arXiv2026

Support Size of $\varepsilon$-Capacity-Achieving Inputs for the Amplitude-Constrained AWGN Channel

We study the amplitude-constrained additive white Gaussian noise (AWGN) channel from the perspective of near-optimal input distributions. While it is known that the capacity-achieving input is discrete with finitely many mass points, the precise scaling of its support size as a function of the amplitude constraint remains an open problem. In this work, we instead consider the minimal support size required to achieve capacity up to an $\varepsilon$-gap. We introduce the quantity $K_\varepsilon(A)$, defined as the smallest support size among discrete inputs supported on $[-A,A]$ that achieves mutual information within $\varepsilon$ of capacity. We show that this relaxed formulation is significantly more tractable and admits sharp characterizations across different regimes of $\varepsilon$. In particular, when $\varepsilon$ decays polynomially with $A$, i.e., $\varepsilon = A^{-β}$ for $β\geq 1$, we establish that $K_\varepsilon(A) = Θ(A\sqrt{\log A})$. For exponentially small gaps, we obtain bounds of order between $A\sqrt{\log A}$ and $A^{3/2}$. Our approach combines approximation-theoretic bounds for Gaussian mixtures with information-theoretic control of entropy via $χ^2$-divergence, together with a wrapping argument that relates the problem to approximating the uniform distribution on the circle. Beyond the technical results, our framework provides a conceptual explanation for the variety of scaling laws observed in prior numerical studies, showing that these correspond to different regimes of $\varepsilon$-optimality rather than intrinsic properties of the exact optimizer.

cs.IT↗

arXiv2026

Explicit Factorization of $x^{p+1}-1$ over $\mathbb{Z}_{p^e}$: A Structural Approach via Dickson Polynomials

Let $p$ be an odd prime. The factorization of the polynomial $x^{p+1}-1$ over the integer residue ring $\mathbb{Z}_{p^e}$ is pivotal for constructing cyclic codes with Hermitian symmetry, a critical resource for Linear Complementary Dual (LCD) codes and Entanglement-Assisted Quantum Error-Correcting Codes (EAQECC). Traditionally, lifting factorizations relies on the generic Hensel's Lemma, masking the underlying algebraic structure. In this paper, we establish a structural isomorphism between the lifting process and the roots of a special auxiliary polynomial $V(x)$, unveiling a deterministic link to Dickson polynomials. Based on this theory, we develop \texttt{Dickson-Engine}, a linear-time algorithm ($O(ep)$) that outperforms standard libraries by orders of magnitude. Applying this engine to $\mathbb{Z}_{169}$, we explicitly construct a family of classical LCD codes of length $n=182$ via the isometric Gray map. Our search reveals codes with parameters (e.g., $[182, 1, 168]_{13}$ and $[182, 2, 144]_{13}$) that are \textbf{near-optimal} with respect to the theoretical Griesmer Bound. Notably, we discover a ``robustness plateau'' starting from non-trivial dimensions ($k=4$), where the minimum distance remains stable ($d=120$) even as the dimension triples ($k=4 \rightarrow 12$). These codes provide exceptional resources for post-quantum cryptography and quantum error correction without entanglement consumption ($c=0$).

cs.IT↗

Compare source metadata on this page

Work	Published	Source identifier	Source
Marker-Delimited Codes for Short-Blocklength, High-Rate Coding over Multi-Read Edit Channels	2026-08-31	2608.30453	arxiv
Intelligent Reflecting Surface Deployment for Low-Altitude Coverage: Illumination Geometry, Directional Characteristics, and Optimization	2026-08-31	2608.30586	arxiv
Reduced polynomial lifts of APN permutations over Galois rings and effective non-APN bounds	2026-10-04	2608.30808	arxiv
Asymptotic Bounds on Generalized Covering Radii of Binary Primitive BCH Codes	2026-08-31	2608.30961	arxiv
Fast Fault-Tolerant Decoders for Hypergraph Product and Lifted-Product Codes	2026-08-31	2608.31040	arxiv
Minimax bounds for watermarked and masked recursive discrete distribution estimation	2026-08-31	2608.31091	arxiv
Semantic Freshness Optimal Sampling and Transmission for Gossiping Receivers	2026-08-31	2608.31140	arxiv
Local Private Information Retrieval for Graph-Based Replicated Systems	2026-08-31	2608.31150	arxiv
Stochastic complexity of vectors containing cluster structure	2026-08-31	2609.00084	arxiv
An Approach to Asynchronous Unsourced Random Access	2026-08-31	2609.00236	arxiv
An Algebraic Framework for Data Systems: Classification and Optimization of Algebraic Structures for Data Organization and Coding	2026-10-12	2609.00273	arxiv
Shannon's problem on the monotonicity of entropy and a Conjecture of Tao	2026-08-31	2609.00399	arxiv
A Public-Key-Dependent Adversarial-Deletion Ceiling for Fixed-Alphabet Multi-Bit Pseudorandom Codes	2026-08-31	2609.02943	arxiv
Learning to Transmit Over Unknown Erasure Channels with Empirical Erasure Rate Feedback	2026-08-30	2507.08599	arxiv
Improving the decoding performance of CA-polar codes	2026-08-30	2512.10223	arxiv
Term Coding: An Entropic Framework for Extremal Combinatorics and the Guessing--Number Sandwich Theorem	2026-08-30	2601.16614	arxiv
Support Size of $\varepsilon$-Capacity-Achieving Inputs for the Amplitude-Constrained AWGN Channel	2026-08-30	2604.14915	arxiv
Explicit Factorization of $x^{p+1}-1$ over $\mathbb{Z}_{p^e}$: A Structural Approach via Dickson Polynomials	2026-08-30	2604.19038	arxiv

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