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At least 19 recordsLinked to original sources

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

On the Power Distance Representation of Bregman Divergences

We express Bregman divergences between parameters as power distances between corresponding weighted points, and investigate several applications of this representation. First, we show that the Bregman closest pair and diameter problems can be reformulated as bichromatic weighted point problems under the power distance. This reformulation enables Bregman nearest and farthest neighbor queries under power distances to be handled in practice through corresponding Euclidean neighbor queries in one additional dimension, after offsetting the weights. Next, we revisit the problem of computing minimum enclosing Bregman balls (Bregman MEBs) and establish their equivalence to double-sided weight power MEBs. We further show that Bregman MEBs can be approximated by relaxing the double-sided weight power MEB problem to its single-sided weight counterpart. We then present two approaches to solving general power MEB problems: An exact algorithm based on the LP-type framework and an arbitrarily accurate approximation scheme based on iterative Frank-Wolfe optimization. Finally, we show that the potential-function lifting transformations used to construct Bregman Voronoi diagrams can be reinterpreted as classical paraboloid lifting transformations applied to the corresponding weighted point sets. This establishes a direct connection between Bregman Voronoi diagrams and power diagrams, further illustrating how Bregman divergences can be represented as power distances.

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

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↗

arXiv2026

The Cross-Correlation Distribution of the Niho-Type Decimation $d=4(2^m-1)+1$

The cross-correlation problem is a classical problem in sequence design. In this paper, we determine the cross-correlation distribution of the Niho-type decimation $d=4(2^m-1)+1$ over $\mathbb F_{2^{2m}}$ for every positive integer $m$. With $q=2^m$, this is equivalent to determining the distribution of the number of roots in $U_{q+1}$ of $f_a(x)=x^7+ax^4+\bar a x^3+1$ for $a\in\mathbb F_{q^2}$, where $U_{q+1}=\{x\in\mathbb F_{q^2}:x^{q+1}=1\}$. The main difficulty is to count the four-element subsets of $U_{q+1}$ that may occur as root sets of $f_a$. We normalize such subsets by their product and study the resulting condition through the associated resolvent. This reduces the required enumeration to several equations over $\mathbb F_q$, which are evaluated by character sums and Kloosterman sums. The remaining mixed Kloosterman sum is related to a Kloosterman sum over $\mathbb F_{2^{2m}}$ and is evaluated using a theorem of Carlitz. Consequently, we obtain explicit formulas for all the frequencies in the cross-correlation distribution.

cs.IT↗

arXiv2026

Embedding Linear Codes into Optimal Self-Orthogonal Codes Based on Constraint-Preserving Genetic Algorithms

We study the problem of finding shortest self-orthogonal embeddings of a given code which have the largest possible minimum distance by employing a constraint-preserving genetic algorithm. We show that the orthogonal group is generated by transvections corresponding to weight-four vectors. Using this result together with the fact that all shortest self-orthogonal embeddings can be obtained through the action of the orthogonal group, we construct a genetic algorithm in which each chromosome is represented by a sequence of weight-four vectors. Applying our algorithm to codes with dimensions $9\le k\le 15$ and lengths $22\le n\le 256$, we obtain 280 inequivalent optimal self-orthogonal codes. In particular, we obtain 34 codes with new parameters and 100 codes which are inequivalent to previously known codes with the same parameters.

cs.IT↗

arXiv2026

Coherent Information Deletion: Bayesian, Generalized Bayesian and Possibilistic Unlearning

Bayes' theorem admits an information processing interpretation due to Zellner (1988): under the Shannon information criterion, the posterior is the unique rule that processes prior and data information without information loss. We revisit these ideas from the perspective of information deletion. Given a posterior based on a complete dataset, what distribution should replace it when a subset of the data is removed? For ordinary Bayesian inference, we show that the corresponding information deletion objective is uniquely minimized by the leave-data-out posterior. We then extend the framework beyond likelihood-based inference from Bayes to the generalized Bayesian updating of Bissiri et al.(2016} based on loss functions. Requiring deletion to be coherent under sequential removal of loss contributions yields a deletion rule that exactly recovers the generalized posterior based only on the retained information. We further develop an axiomatic theory, beyond probabilistic inference, for possibilistic deletion and show that the same information-deletion perspective leads to a principled rule for exact possibilistic unlearning. Finally, restricting these exact deletion principles to tractable families leads naturally to variational Bayesian, generalized Bayesian, and maxitive possibilistic unlearning.

stat.ME↗

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

Adaptive Beam Hopping and Power Control for Dual-Layer Over-the-Air Online Federated Learning in LEO Satellite Networks

This paper investigates over-the-air (OTA) computation enabled online federated learning (FL) in low-Earth orbit (LEO) satellite networks. Specifically, we consider a dual-layer OTA aggregation architecture, where ground devices upload analog model updates to serving satellites via uplink OTA aggregation, and satellites forward the aggregated signals to a data processing center through the second round OTA aggregation. Then, we formulate a long-term data-utilization maximization problem in which devices continuously collect new data and untrained samples gradually lose freshness. The problem is subject to the satellite beam budget, transmit-power limit, and global mean squared error (MSE) constraint that governs end-to-end aggregation distortion. This yields a coupled mixed-integer nonlinear programming (MINLP) problem, involving tightly coupled discrete beam-hopping decisions and continuous power control. Due to the combinatorial action space and nonconvex constraints, the problem is NP-hard and computationally intractable. Furthermore, the time-varying satellite topology and dynamic data generation render it a sequential decision-making problem, necessitating adaptive online scheduling. To address these issues, we cast the problem as a Markov decision process and develop a proximal policy optimization (PPO)-based deep reinforcement learning framework that jointly optimizes adaptive beam hopping and power control, using an MSE-aware reward to balance data utilization and aggregation accuracy. Numerical simulation results verify that the proposed algorithm consistently outperforms other benchmark schemes, achieving superior long-term data utilization and faster FL convergence while satisfying the MSE requirement.

cs.IT↗

arXiv2026

An Identifiability Theory of Masked Prediction: Mode Blindness and Mask Schedules

Masked prediction learns to infer missing variables from visible context selected by a mask schedule. When does small excess risk guarantee recovery of the true joint distribution? We study this question on finite product spaces under masked-block log loss, with conditionals induced by a single joint distribution. To quantify recovery, we introduce an $\varepsilon$-identifiability modulus measuring the worst-case error among joint distributions with population excess risk at most $\varepsilon$. For data with separated modes pinned down by sufficiently large visible contexts, schedules that always retain such contexts can permit nonvanishing mode-weight errors while incurring exponentially small excess risk. An information decomposition explains why: the loss captures mode-weight mismatch only where the visible context leaves the mode uncertain. We prove two-sided bounds showing that, over a fixed range of mode weights, sensitivity to mode reweighting is governed by residual mode uncertainty averaged over the mask schedule. Assigning schedule mass to low-visibility masks that retain this uncertainty yields recovery bounds within the reweighting family. Beyond this family, positive full-mask probability characterizes uniform control of joint KL divergence by excess risk. Exact calculations and controlled gradient optimization validate these predictions.

cs.LG↗

arXiv2026

Quantum Message Passing Convergence and Vanishing Block-Error Probability for Random LDPC Codes

Belief propagation with quantum messages (BPQM) is a quantum algorithm that decodes classical codes transmitted over classical--quantum channels. It realizes optimal decoding on tree factor graphs over pure-state classical-quantum channels. However, this tree-based analysis does not ensure vanishing block-error probability for LDPC Tanner graphs with cycles. In this work, we construct a two-stage BPQM decoder for random $q$-ary LDPC codes over symmetric $q$-ary pure-state channels, where $q$ is prime, and prove that its ensemble-average block-error probability vanishes as the blocklength $N$ tends to infinity. For regular ensembles with $d_v\geq3$, fidelity bounds yield double-exponential decay of the average symbol-error probability throughout the BPQM success region. We apply depth-$\ell$ BPQM to coordinates with tree neighbourhoods and treat the remaining coordinates as erasures. With a suitable $\ell=Θ(\log\log N)$, a noncommutative union bound controls the BPQM decoding errors, while the minimum-distance property guarantees erasure recovery. We also extend the analysis to finite-support irregular ensembles. These results are relevant to quantum algorithms based on Regev's reduction, where coherent decoding uncomputes a codeword register. Decoded quantum interferometry (DQI) uses a closely related Fourier-based framework that reduces sparse max-LINSAT optimization problems to LDPC decoding problems on pure-state channels. Our results justify the use of BPQM in the decoding step of DQI and of coding-theoretic algorithms based on Regev's reduction whenever the code is drawn from one of the random LDPC ensembles analyzed here and the induced memoryless symmetric pure-state channel lies in the BPQM success region. Curiously, our numerical results indicate that DQI+BPQM achieves a satisfaction ratio that closely matches that of simulated annealing.

quant-ph↗

arXiv2026

Answer-Distribution Trajectories: A Stochastic-Dynamics View of LLM Reasoning

Chain-of-thought reasoning provides a structured computation between a model's input and final answer. Yet it is often evaluated through endpoint accuracy, which ignores the path taken to reach that answer. An emerging line of work addresses this limitation using entropy profiles, which track how uncertainty evolves over the reasoning process but do not reveal which competing hypotheses account for that uncertainty. We introduce answer-distribution trajectories, a stochastic-dynamics-inspired representation that tracks the model's full predictive distribution over answers as reasoning unfolds. As a strictly finer representation than endpoint and entropy summaries, answer-distribution trajectories enable us to characterize a trace through a dynamical reasoning profile spanning exploration, revision, motion, and commitment, and to distinguish different dynamical mechanisms of reasoning success and failure. Across sixteen open-weight language models and four reasoning benchmarks, we show that traces with the same endpoint and similar entropy profiles can exhibit substantially different reasoning dynamics. We further find substantial variation in these dynamics both within and across models and tasks, with different objectives favoring different dynamical profiles. Additionally, we show that training and inference choices systematically reshape these profiles. Our results suggest that answer-distribution trajectories provide a rich framework for analysing and evaluating the dynamics of LLM reasoning.

cs.AI↗

arXiv2026

Universal recovery in approximate quantum error correction

Universal recovery -- the existence of a single recovery map that corrects an entire family of error channels -- is a central feature of quantum error correction (QEC). In exact QEC, linearity guarantees that a code correcting a given error set also corrects every channel whose Kraus operators lie in its linear span, and that a single recovery map suffices for all such channels. Approximate quantum error correction (AQEC), which relaxes perfect recovery to recovery with controlled error, has traditionally lacked this structure. In a recent paper (arXiv:2607.22995), we developed a theory of approximate quantum error correction showing that a restricted form of linearity persists in the approximate setting, yielding uniform AQEC guarantees for the family of channels controlled by a given error set. In this work, we complete the picture by establishing the second half of universal recovery in the approximate setting: a single recovery map can simultaneously correct every channel controlled by a given error set. The error-set theory we proposed quantifies approximate correctability through two parameters: the environment-leakage distance, governing worst-case performance, and the Knill--Laflamme Hellinger distance, governing average-case performance. We show here that both quantities also control universal decoding. We further study the Petz map naturally associated with an error set as an explicit universal recovery, and obtain uniform average- and worst-case guarantees across the entire family of channels.

quant-ph↗

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

Degree Sequence Reconstruction from Subgraph Traces

The goal of degree sequence reconstruction is to recover the ordered vector of degrees of an unknown graph from vertex deleted traces, where each vertex is deleted independently with probability $p$. We provide two algorithms for reconstruction; the first uses rejection sampling to reduce the problem to an estimation problem for a mixture distribution. Combined with prior trace reconstruction results, this gives a reconstruction algorithm using $\Exp{\tilde O (n^{1/3})}$ traces, although no sub-exponential time decoder is known. Our other approach involves recovering certain graph invariants, degree moments, that can identify a graphs degree sequence. Extremal polynomial bounds show that $\tilde Θ(n^{1/2})$ degree moments are necessary and sufficient to reconstruct the degree sequence, which leads to an algorithm with $\Exp{\tilde O(n^{1/2})}$ trace complexity. The same polynomial machinery yields a sub-exponential time decoder for the degree sequence from the moments. Additionally, we give an $O(n^{3})$ upper bound and a $Ω(n^2)$ lower bound for the trace complexity of recovering the number of edges.

cs.IT↗

arXiv2026

Condense to Conduct and Conduct to Condense

In this paper, we present the first explicit examples of low-conductance permutations. The notion of conductance of permutations was introduced by Dodis et al. in "Indifferentiability of Confusion-Diffusion Networks", where the search for low-conductance permutations was first initiated and motivated. As part of our contribution, we not only provide these examples, but also offer a general characterization of the problem: we show that low-conductance permutations are equivalent to permutations possessing the information-theoretic properties of Multi-Source-Somewhere-Condensers, a specific variant of somewhere condensers.

cs.CR↗

arXiv2026

Binary Multiple-Node-Erasure-Correcting Codes over Complete Graphs: Constructions, q-Ary Metric Balls, and Duality

We study linear codes whose coordinates are the ordinary edges and self-loops of complete undirected graphs; a node erasure removes all coordinates incident with a failed vertex. The construction results are binary. For triple-node erasures, we extend the published cyclic construction by allowing a suitable cyclic check slope to depend on the prime graph length. An explicit determinant test proves that one of three fixed slope choices works at infinitely many prime lengths, unconditionally, and gives redundancy $3n-2$, one bit above the graph Singleton bound. We also give Singleton-optimal triple-node codes at $n=6,8,10,12$, together with a general ordinary-edge framework that isolates the remaining loop-completion problem. When $2$ is primitive modulo an odd prime $n$, a binary multi-slope construction corrects every $ρ$-node erasure for $2\leqρ<n$, with redundancy $ρn-(ρ-1)$ in the range $2\leqρ\leq(n+1)/2$. Returning to arbitrary prime powers, we derive exact generating transforms and inclusion--exclusion formulas for node-metric ball volumes, fixed-radius asymptotics, and packing, existence, and covering bounds. Finally, for the complementary clique-erasure metric, we obtain an exact weight enumerator and a Singleton-optimal node--clique duality.

cs.IT↗

arXiv2026

The half-rate linear programming bound for binary codes is $\frac12-\frac1π$

In their work on sphere packing and the modular bootstrap, Afkhami-Jeddi, Cohn, Hartman, de Laat, and Tajdini conjectured the exact high-dimensional exponent of the Cohn-Elkies sphere-packing linear program. OpenAI's Chapter 1 subsequently proved their conjecture by establishing that both Fourier sign-uncertainty radii are $(1/π+o(1))\sqrt d$. We prove the binary coding analogue: \[ R_D\!\left(\frac12-\frac1π\right)=\frac12. \] We also formulate the two Krawtchouk sign-uncertainty problems and determine both of their asymptotics. If $A^{\mathrm K}_{\pm}(n)$ denotes the smallest radius $r$ for which a nonzero Krawtchouk $(\pm1)$-eigenfunction $f$ exists with $f(0) = 0$ and $f(x) \ge 0$ for all $|x| \ge r$, then \[ \frac{A^{\mathrm K}_{\pm}(n)}n\longrightarrow \frac12-\frac1π. \] The lower bound proves a mass-concentration principle similar to OpenAI's Chapter 1 for Hamming space. The upper bound, on the other hand, follows the approach of the spherical-code construction in OpenAI's Chapter 2. Gay, Jeronimo, and Liu formulated a hierarchy for binary codes analogous to the spherical-code construction and improved the best known binary coding rate bounds by evaluating the first level of the corresponding hierarchy. We prove this hierarchy bounds the Delsarte program and give a construction at arbitrarily deep levels of the hierarchy, attaining the upper bound in the limit. The construction uses an $N$-qubit generalization of the pure-state channel of Alrabiah and Guruswami.

math.CO↗