ALT Maximum rank-distance codes were introduced independently by Delsarte (1978) and Gabidulin (1985) for unrestricted matrix spaces, and their study was subsequently extended to the classical form spaces of alternating, symmetric, and Hermitian matrices. For Hermitian d-codes, Schmidt (2018) established the upper bound and constructed optimal additive codes for all admissible parameters except when n and d are both even and 4≤ d≤ n-2. We settle this final open existence case, more generally, construct additive maximum Hermitian rank-metric codes for every admissible length and minimum distance. Rather than working directly in the space of Hermitian matrices, we construct our objects within central simple algebras with unitary involution arising from quotients of skew polynomial rings. We also introduce scalar-field and Lyapunov operators in these algebras, leading to new invariants for the equivalence of additive Hermitian codes. Explicitly determining these invariants yields strong neces

ALT We introduce and study the adaptive covering complexity C adapt S (s), defined as the minimum number of contiguous factor blocks required to partition a string s when each block is either a factor of a fixed external dictionary S or a factor of the target's already decoded prefix. This hybrid admissibility model formalizes the structural interaction between static dictionary matching and dynamic nonoverlapping self-referencing. Our main theoretical result is the exact logarithmic law which is tight and proves that dynamic self-reference can reduce phrase count by at most an exponential factor over static dictionary covering alone. We establish structural equivalences showing that the measure bridges word-segmented Relative Lempel-Ziv and non-overlapping LZ77, and we prove exact preset-history LZ77 sandwich bounds. Furthermore, we establish the optimality of the longest-match greedy parse via an exchange lemma and present an optimal implementation based on dual suffix automata running i

ALT In this paper, we investigate the information bottleneck problem for sources that follow discrete-time ergodic finite-state Markov chains. We propose treating observations generated by the time-varying Markov chain as if they were drawn from its unique stationary distribution, and we use the solution to the information bottleneck problem for the stationary process to compress samples from the time-varying process. The resulting error is upper bounded by several information gap terms. We establish sufficient conditions under which these gaps converge to zero and analyze the convergence rates of the different information gap terms. The proofs are based on intricate analysis using convergence of subsequences along with the derivation of properties of the optimization sets associated with the information bottleneck problem. In addition to the inequalities and convergence results we establish, our analysis offers insights into how simple properties of ergodic Markov chains give rise to resu

ALT We propose a finite BSC-decomposition method for representing an arbitrary binary-input symmetric-output channel (BISOC) as a mixture of binary symmetric channels (BSCs) with at most ℓ+1 distinct crossover probabilities. We show that the weighted average capacity of the component BSCs exceeds the capacity of the original BISOC by at most (π²log e)/(48ℓ²)+O(ℓ⁻⁵/²) bits, and the leading term is asymptotically optimal in the worst case. Based on the finite BSC-decomposition, we design a channel simulation scheme for arbitrary BISOCs. Combined with efficient channel simulation algorithms for BSCs, our scheme achieves a communication rate with O((log n / n)²/³) redundancy over the capacity for simulating n copies of a BISOC.

ALT Secure aggregation enables a server to compute the sum of users' gradients or model updates without revealing individual values. With user dropouts, correlated random keys protect local gradients while allowing recovery of the desired gradient sum. We consider three phases: key distribution, masked-gradient upload, and mask elimination. Users establish correlated keys, upload masked gradients, and provide additional messages to eliminate residual aggregate masks. Existing information-theoretic schemes typically design mask-elimination communication for a prescribed dropout threshold and evaluate its cost under worst-case dropouts across the masked-gradient upload and mask-elimination phases. Such designs cannot adjust communication to actual dropout scenarios. Moreover, the randomness and communication costs of establishing correlated keys remain largely unexplored. To address these limitations, we formulate an adaptive secure aggregation framework under the T-colluding security constr

ALT We propose a posterior-mass rescue (PMR)-aided shifted-pruning SCL-flip decoder for CRC-aided polar codes. After CRC-aided SCL (CA-SCL) and differential-metric shifted-pruning recovery fail to find a CRC-valid candidate, PMR locally continues discarded prefixes stored during the initial CA-SCL pass, ranked by estimated posterior mass, without increasing the shifted-pruning retry budget. Simulation results for a (128, 64) code with the Reed-Muller (RM) rate profile and a (256, 128) code with the polarization-weight (PW) rate profile, both using an 11-bit CRC, show lower observed FER than SCLF-Diff (T = 15) and the considered literature baselines under the evaluated recovery budgets, with observed FER close to that of SCLF-Diff (T = 20). At the reported moderate and high SNRs, the average reconstruction-aware path-step count is within a few percent of that of SCLF-Diff (T = 20), while at 2.0 dB it is 9.4% and 4.3% higher for the two codes, respectively. The proposed decoder also has a la

ALT Recent advances in artificial intelligence have contributed to several information-theoretic results through genuinely different modes: a language model may suggest a proof, participate in an extended mathematical dialogue, coordinate computation, or operate as one component of a search harness. Treating all of these as "AI-discovered mathematics" obscures distinctions that matter for scientific reliability, attribution, and reproducibility. This article examines a set of recent cases through a four-stage pipeline: how the problem was formulated; how AI participated in finding a solution; how the resulting analytical and computational claims were verified; and how machine-generated outputs were translated into mathematical exposition for human understanding. Based on these cases, we recommend that research publications (1) disclose the nature and extent of AI assistance at each stage, (2) report verification of analytical reasoning separately from certification of numerical computation

ALT An agent can retain information before receiving a future question and later retrieve evidence from an archive. Evaluating this arrangement requires accounting for both retained information and the width of each retrieval response. Decision-tree capacity bounds describe individual sets of distinguishable records, but a memory writer must encode every possible record before the query is known. We study fixed-length retention beside an immutable binary archive, with a worst-case limit on raw bit or word reads. We characterize feasible one-bit-read memory cells and construct encoders that partition the complete record space. Supporting pairwise exclusive disjunction and conjunction with one bit read is equivalent to supporting every Boolean query. For multiple reads, we combine explicit encoders with decision-tree counting and an entropy converse. The word-read bounds differ by at most two retained bits when the read budget is at most half the archive length. Under an explicit condition o

ALT How much mutual information is needed to achieve a target correlation between two copies of a prescribed distribution? We prove a sharp Gaussian extremal inequality for this self-coupling problem: among distributions with a fixed mean and covariance, the Gaussian has the greatest minimum information cost for meeting a positive semidefinite cross-covariance target. The inequality is strict for non-Gaussian distributions when the target is positive definite. The proof uses a midpoint gluing construction. Two Ornstein–Uhlenbeck half-bridges, each started from the prescribed endpoint law, meet at a common intermediate distribution and are joined by conditional gluing. Changing the Gaussian reference then turns the resulting relative-entropy bound into the desired extremal inequality. As a first application, scaling one endpoint of the self-coupling establishes Gaussian extremality for rate-distortion-perception with unlimited common randomness, under quadratic distortion and squared Wasser

ALT Denoising diffusion probabilistic models (DDPMs) have emerged as a powerful framework for generative modeling by transforming standard Gaussian inputs into outputs that approximate a target distribution. While DDPMs have been proved to converge in total variation, KL divergence, and Wasserstein distance, these discrepancy measures share a fundamental limitation of being insensitive to distributional mismatch in low-probability regions. To address this, we adopt the stronger Rényi divergence as the discrepancy measure. Leveraging a Girsanov-type change-of-measure argument, we establish the first Rényi characterization of DDPMs to the best of our knowledge, showing that the total error decomposes into a mixing error, a discretization error, and a score estimation error. We assume the target distribution is sub-Gaussian and characterize the score error via the cumulant generating function, relaxing both the log-Sobolev-inequality and L∞-score-error conditions used in prior Rényi divergenc

ALT We study quantization for inner-product estimation when storage and reading have separate budgets. In the static bit-probe model [Yao, 1981], an encoder stores w bits about x before seeing a query y, and a reader probes at most r<w bits to estimate x⊤ y. Both vectors lie in the d-dimensional Euclidean unit ball. Protocols may be randomized, and computation is unrestricted. Performance is measured by d times the worst-case mean-squared error over x,y, averaged over protocol randomness. First, we show that query-independent reading cannot benefit from extra storage. Any such reader is equivalent to fully reading an r-bit message, even when later reads depend on previously returned bits. At fixed positive rates W=w/d and R=r/d, the normalized minimax error converges to 1/(4ᴿ-1), independent of W. The upper bound was already known [Ordentlich and Polyanskiy, 2025]. We prove the matching lower bound for arbitrary scalar decoders, extending the result for unbiased compressors [Safaryan et al