Papers

We study joint recovery of the load (the number of active users) and message-count vector in type-based unsourced random access with very short messages. Each user's $k$-bit message activates one of $M=2^k$ orthogonal resources; coherent collisions therefore produce a noisy, unnormalized histogram of the transmitted messages. For arbitrary load and message priors, we propose an exact joint maximum-a-posteriori (MAP) decoder. Known-load allocation is separable and convex; our contribution is to show that the solutions for all candidate loads are nested prefixes of one greedy chain. Scoring these prefixes with the Bayesian posterior terms jointly estimates the load and message-count vector in $O(M+K_{\max}\log M)$ time. We compare the proposed joint MAP decoder with marginal MAP, three scalar-count two-stage receivers, oracle-load MAP, a sum-only receiver, and a soft-pilot receiver. For $K\sim\operatorname{Unif}\{0{:}60\}$ at 2 dB, the proposed decoder cuts the no-activity false-alarm probability from $25.9\%$ to $3.5\%$ relative to the Bayesian-count two-stage receiver and, conditioned on $K\ge1$, reduces normalized count error by $21.4\%$. The advantage concentrates under idle or sparse uncertain activity, whereas support-count processing approaches joint MAP when the possible loads are well separated. We also prove that the sum-only statistic is insufficient and quantify prior and amplitude mismatch. The model targets synchronized, power-controlled, very-short payloads rather than general URA.

We study cache-aided linear computation in distributed systems where each source generates and stores its own data. An external receiver requests a linear combination of these data, and the sources use information cached before the request is known to reduce the communication needed to recover the requested result. In our model, each of the $N$ sources stores its own independent file over a finite field and has an additional cache of normalized size $M$. The receiver must recover any requested linear combination with zero error from the messages sent by the sources over separate noiseless links. We establish a lower bound on the worst-case communication rate over any finite field by analyzing several requests under the same cache placement. This bound determines the optimal rate when no additional cache is available. For binary files, we propose parity-aided placement (PAP), a coded-placement scheme that combines cached parities with uncoded symbols. For every $N\geq3$, we characterize its communication rate for each request and optimize its placement parameters to obtain a closed-form worst-case rate. The resulting achievable rate meets the lower bound for $0\leq M\leq1/(2N-1)$ and $M\geq(N-1)/N$, establishing the optimal memory–rate tradeoff in these two ranges.

Spatial reception under directed molecule transport requires balancing useful molecule capture against intersymbol interference (ISI). To characterize this balance, we study a fully absorbing spherical receiver that counts absorption events only within a prescribed cap. Using the known joint first-hitting law under aligned drift, we derive exact cap impulse, cumulative, and eventual responses. We perform stable cumulative evaluation. We then assess the derived responses through particle simulations and time-step comparisons, and demonstrate that our conditional-timing analysis explains when opposite drifts of equal magnitude can favor the same wider aperture. We also show that for the studied geometry and a symbol duration of $0.15$ s, the aperture that maximizes the signal-to-interference difference (SID) increases from $29.13^\circ$ at zero drift to $61.86^\circ$ at a drift magnitude of $40$ $\mu$m/s. These results (i) connect joint arrival-time and location statistics to spatial reception design, (ii) characterize the tradeoff between useful molecule capture and ISI, (iii) support SID-based aperture selection for a given drift and symbol duration, and (iv) quantify the SID loss incurred by retaining the zero-drift optimal aperture for the same symbol duration.

We study Boolean entropy and Fisher information in information inequalities and the Boolean central limit theorem. We prove a fractional Stam inequality for the Boolean Fisher information associated with arbitrary fractional partitions of subset sums, and derive an entropy power inequality for sums of Boolean independent random variables and its weighted version. We then establish a Boolean analogue of Barron's entropy convergence result along the Boolean central limit theorem: Boolean entropy converges to the entropy of the Rademacher law if and only if the initial entropy is finite. Furthermore, under a finite fourth moment, in the Boolean central limit theorem, we obtain \[ \Gamma(\mu_n\mid\mathbf b)= \frac{m_4(\mu)-1-m_3(\mu)^2}{2}\frac{\log n}{n} +o\!\left(\frac{\log n}{n}\right), \] with an explicit example showing sharpness. For symmetric laws with finite fourth moment and finite Boolean Fisher information, \[ \Psi(\mu_n\mid\mathbf b) =2\bigl(m_4(\mu)-1\bigr)\frac{\log n}{n} +o\left(\frac{\log n}{n}\right). \] Our approach is based on the Nevanlinna representation of Cauchy transforms.

We introduce binary cyclic gradient coding (BCGC), an exact gradient coding scheme for $n$ workers and up to $s$ stragglers. Its encoding matrix is binary, circulant when $(s+1)\mid n$ and nearly circulant otherwise, and the master decodes by a single summation. Like the fractional repetition coding scheme (FRC) and its generalized version (GFRC), BCGC splits the workers into $s+1$ classes that each cover the whole dataset, but it assigns the data through cyclic supports. Its encoding matrix has rank $n-s$, and $n-s-1$ workers keep the balanced load $s+1$, which is the largest possible number. We prove that encoding matrix of BCGC is not permutation-equivalent to those of FRC and GFRC schemes. Moreover, We also prove that a binary coded scheme with compression is equivalent to an uncoded compressed scheme on a coarser partition of the data. This explains why binary schemes remain stable under compression, whereas the large decoding coefficients of MDS codes amplify the compression error and lead to divergence in our experiments. For every contractive compressor, the standard convergence guarantee of compressed gradient descent holds on each coarse partition. FRC, GFRC and BCGC therefore share the same convergence guarantee and differ only in their partitions, with one partition for FRC, two for GFRC and $s+1$ distinct partitions for BCGC. Choosing among these partitions can recover the exact gradient in cases where FRC and GFRC cannot. Experiments on four datasets confirm the theory. BCGC matches GFRC, keeps its accuracy under top $\kappa$ sparsification while the MDS codes diverge, and on heterogeneous data its partition diversity reduces the compression error and improves the accuracy.

Network coding allows intermediate nodes to combine messages, whereas routing only forwards them. The Li-Li conjecture asserts that network coding and fractional routing have the same capacity region in undirected multiple-unicast networks. We give a counterexample to this conjecture and apply it to external-memory integer sorting. Since Aggarwal and Vitter posed the question in 1988, whether integer sorting can beat the external-memory comparison bound has remained a central and extensively studied open problem. We give a deterministic sorting algorithm: for $N$ integers of $\Theta(\log N)$ bits, a memory holding $M$ integers, and block transfers of $B$ integers, where $2B\le M\le N$, the number of block transfers is $O\left(\frac{N}{B}\left[1+\frac{\log N}{(\log(M/B))^{1-\varepsilon}(\log M)^\varepsilon}\right]\right)$, where $\varepsilon>0$ is an absolute constant. In a range of parameter regimes, this bound is asymptotically smaller than the classical comparison-sorting bound. We also use Lean 4 to verify the sorting algorithm's correctness and complexity bounds, and the strict advantage of coding over routing.

Let $A_q(n,d)$ denote the largest size of a code of length $n$ and minimum Hamming distance $d$ over an alphabet of $q$ symbols. For fixed $q\ge2$ and $d\ge4$ we construct codes with redundancy $n-\log_q A_q(n,d)\le(d-3)\log_q n+O(\sqrt{\log n})$ for all large $n$. No algebraic structure is imposed on the alphabet, and the codes are not linear. For $d=4$ and $d=5$ the coefficient of $\log_q n$ matches the Hamming bound, so the redundancy is determined up to lower-order terms. At distance four the codes are provably better than every linear code: by the polynomial-method bound for cap sets, $\mathbb{F}_q$-linear codes with $q>2$ require a strictly larger coefficient in front of $\log_q n$. The construction imposes moment checksums over an auxiliary finite field, as BCH codes do, but one fewer than the distance would normally require. The affine-line theorem from prior work describes the dependencies left by these checks. We exclude them by placing the code positions inside a specially structured subset of the field, built from Behrend's sphere construction with the help of algebraic number theory.

Bilinear Compressive Security (BCS) is a lightweight encryption scheme that combines compressed sensing with a fresh random convolution filter to improve resistance against key-recovery attacks under key reuse. Previous work derived information-theoretic guarantees for BCS by relating key identifiability to phase retrieval under the assumption that an adversary has access to the full ciphertext distribution induced by the random filter. In practice, however, an adversary only observes finitely many encryptions, leaving the security of BCS in the finite-sample regime largely unexplored. In this paper, we study key recovery from only finitely many observations. We first develop a covariance-based attack that estimates the linear measurements from repeated encryptions of the same plaintext and subsequently recovers the secret measurement matrix. We analyze the estimation procedure and derive finite-sample guarantees together with corresponding sample-complexity bounds. We further investigate the complete attack experimentally and compare its observed recovery behavior with the previously established information-theoretic identifiability results. Our experiments show two qualitatively different recovery regimes on the structure of the measurement matrix. For dense matrices, including Gaussian matrices, the number of measurements required for successful recovery appears to grow linearly with the signal dimension. This differs from the worst-case behavior suggested by information theory. In contrast, for matrices with sparse rows, we observe a quadratic growth consistent with the theoretical bound. These results suggest that the sparsity structure of the measurement matrix plays an important role in the security of BCS and should be taken into account when selecting parameters and designing practical instances of the scheme.

Shannon entropy is a standard measure of ciphertext randomness, evaluated either globally over the byte histogram of the entire ciphertext or locally as the average over non-overlapping blocks of ?xed size. While the global statistic captures only the overall byte distribution, the local statistic is also sensitive to non-randomness that is con?ned to parts of the ciphertext. Both measures are commonly assessed against reference values derived under the assumption of independent uniform bytes. However, STL encodes most numerical data as single-precision IEEE-754 ?oats, and format validity requires them to be ?nite, causing nonuniform byte probabilities and dependence between the two bytes of each exponent pair. We derive the exact mean and variance of entropy under the valid-STL (VSTL) distribution, which assigns equal probability to every admissible sequence of triangle records, and provide second-order expansions that keep the global reference values practical to compute at large ?le sizes. Using these expansions, we establish a two-sided hypothesis test for the global entropy statistic, whose limiting value is lowered from 8 to 7.99991479609 bits. The format constraints also change the local entropy reference: applying the capacity energy criterion to triangle records selects a block size of 1950 bytes instead of 1936 bytes, with an exact reference mean of 7.903106732975 rather than 7.902469316647 bits. Although small, these shifts are su?cient to place the VSTL reference means outside the conventional 95% acceptance intervals at practical ?le sizes, making unadjusted entropy tests prone to falsely rejecting ideal VSTL ciphertexts. Monte-Carlo simulations con?rm all reference values within sampling error. We therefore recommend using the format-speci?c reference values derived here to evaluate STL ciphers.

We consider a discrete memoryless state-dependent channel with noiseless feedback, where the transmitter uses randomized encoding to jointly perform message transmission, identification, and state sensing. In this setting, the transmitter sends a conventional transmission message and an identification message while estimating the unknown channel state sequence under an average distortion constraint. We establish a single-letter characterization of the randomized identification-transmission-sensing capacity-distortion region. The result reveals a single-letter separation structure: the communication rates are determined by the averaged channel, subject to the sensing-induced constraint on admissible input distributions. Specifically, the transmission rate is characterized by mutual information, while the identification rate is characterized by output entropy under the same distortion constraint. The achievability proof combines constant-composition transmission coding, output partitioning, and combinatorial identification coding, while the converse follows from an averaged-channel reduction, distortion lower bounds, and feedback-identification converse arguments.

Accurate information-theoretic analysis of multiuser wavelength-division multiplexed (WDM) optical links is complicated by Kerr nonlinearity and chromatic dispersion. Existing capacity bounds for first-order regular perturbative (RP) channel models have relied on per-symbol peak power constraints to simplify the underlying extremal analysis. In this paper, we replace these peak constraints with the more practically relevant average input power constraints and study the resulting RP WDM interference channel. Unlike the peak-constrained formulation, an exact finite signal-to-noise ratio (SNR) analysis under average power leads in general to a nonconvex, high-dimensional optimization problem with no closed-form solution. We show that the problem admits a tractable characterization in the high-SNR regime. Using a genie-aided argument, we derive closed-form single-user high-SNR capacity bounds for the RP interference channel and identify constant-amplitude interferer signaling as extremal. Furthermore, under an empirically validated non-negativity condition on the dominant RP kernels and within a truncated first-order RP model, we establish a finite-SNR extremality result, yielding a single-user capacity characterization within the RP model under average power constraints. Building on these single-user corner points, we construct an all-helpsone achievable region via time sharing, which strictly improves upon treating nonlinear interference as noise. Numerical results and split-step Fourier simulations of the nonlinear Schrodinger equation channel confirm that the qualitative conclusions persist over practical launch powers. The results provide new insight into extremal signaling and multiuser trade-offs for nonlinear WDM systems within the RP approximation.

Subspace-design codes form a central unifying framework in coding theory, connecting probabilistic existence results and explicit algebraically structured constructions. The two core families are folded Reed-Solomon codes and univariate multiplicity codes, both possessing the subspace-design property. These codes reach list-decoding capacity with polynomial-size lists, but their alphabet size grows with code length and hurts practical efficiency. Goyal, Guruswami, and Hsieh achieved a major breakthrough: they constructed the first explicit subspace-design codes over constant-size alphabets via the Alon-Edmonds-Luby expander framework, settling an open problem and showing subspace-design properties can hold without large alphabets. Despite recent progress, constructing subspace-design codes with low-complexity efficient encoding remains a critical open challenge. This work gives an explicit construction of r-dimensional subspace-design codes over a constant alphabet whose size grows exponentially with r. By comparison, Goyal et al. (FOCS 2026) built r-dimensional subspace-design codes with alphabet size doubly exponential in r squared. Our construction is computationally efficient: encoding scales quadratically with block length and linearly with dimension r in field operations. We use modified algebraic geometry codes, replacing rational evaluation points with higher-degree places to obtain a valid additive code. To confirm parameter bounds, we use Moore matrices to analyze determinants with many zero entries. We also generalize univariate multiplicity codes, originally built from Reed-Solomon codes, to algebraic geometry codes, and prove these AG-multiplicity codes achieve optimal parameters.

Feel-good Thompson sampling (FGTS) augments the posterior of vanilla Thompson sampling with an exploration term that favors reward functions with high predicted optimal values. This modification achieves minimax-optimal regret up to logarithmic factors in linear contextual bandits, but existing analyses of its two standard variants establish regret guarantees in expectation. In this paper, we derive the first high-probability regret bounds for both cumulative and current-step FGTS in contextual bandits with general realizable function approximation. Our bounds depend on the corresponding decoupling coefficients and the prior mass near the true reward function. In the $d$-dimensional linear setting, both variants satisfy $\mathrm{Regret}(T)=O\big(d\sqrt T\log(2T)+\sqrt T\log(1/\delta)\big)$ with probability at least $1-\delta$, where $T$ is the time horizon. This bound matches the dimension and horizon dependence of algorithms based on the upper confidence bounds, up to logarithmic factors. Relative to the expected-regret guarantees, the cost of controlling the regret with high probability is confined to additive terms with at most linear dependence on $\log(1/\delta)$. The proof ideas are partially developed by GPT-6 Astra under the authors' guidance.

Digital-domain self-interference cancellation is essential in in-band full-duplex radios after passive suppression and analog cancellation. It is particularly challenging when the desired signal is strong relative to the residual self-interference. In this regime, conventional iterative digital cancellers update their coefficients from desired-signal-contaminated received samples and may suffer substantial performance degradation. This paper reformulates digital-domain self-interference cancellation as an operator-learning problem. The proposed framework learns the temporal evolution of the residual self-interference channel from historical observations and uses the learned operator together with the known local transmit signal to reconstruct the current residual self-interference. A Transformer neural operator is used to realize this framework. The resulting method is trained in a desired-signal-free setting and then applied in the presence of a desired signal. A continuity-based analysis characterizes the sensitivity of the learned operator to the resulting input perturbation. Numerical experiments in static and time-varying channels, including time-varying SIR conditions, demonstrate stable cancellation performance over a wide range of interference levels. At 30 dB SIR, the proposed method improves the cancellation performance by 5–15 dB over the considered conventional methods.

The problem of Private Information Retrieval (PIR) across graph-based replicated systems has seen a surge of interest in recent literature. In such systems, servers are modeled as the vertices of a graph, and each file is represented by an edge connecting the pair of servers that store it. While previous research on Graph-PIR (G-PIR) has focused mostly on perfect information-theoretic privacy, relaxing this constraint has the potential for significantly higher retrieval rates. In this work, we consider the setting of Graph-based Weak Private Information Retrieval (G-WPIR), under an $\epsilon$-privacy metric. We adapt an existing G-WPIR scheme for arbitrary graphs to analyze its performance under the $\epsilon$-privacy metric. Additionally, we establish a lower bound on the expected download cost for arbitrary graphs under the $\epsilon$-privacy metric and derive from it an explicit rate upper bound for complete graphs. Finally, we show the achievable rate-privacy trade-offs for two specific graph classes—complete graphs and complete bipartite graphs.

We study two explicit constructions of binary Calderbank–Shor–Steane quantum low-density parity-check (QLDPC) codes with check weight six. The first uses a fixed circulant shift pattern and yields parameters $\code{6N}{2N+2}{6}$ with Tanner graph girth twelve for $N=31$ and every $N\ge37$. A graph incidence interpretation gives the ranks and girths, and a uniform weight six logical operator establishes the exact distance. The second construction uses fixed matrix polynomials with circulant and affine permutation coefficients. Here, a polynomial commutation identity guarantees orthogonality for every cyclic lift. Four realizations give $[[192,16,8]]$, $[[288,16,12]]$, $[[384,16,14]]$, and $[[576,32,20]]$, with Tanner girth eight and computationally certified distances. We give reproducible construction data, distinguish classical cycle distance from quantum distance, and identify the limitations of both families.

The torn paper channel breaks a codeword into fragments and returns all fragments without their original order. The noisy torn paper channel additionally introduces bit substitution errors to each fragment. Recovering the message therefore requires accounting for both bit errors and uncertainty about the fragment order. We establish a capacity upper bound for the noisy torn paper channel that matches the achievable rate of Ravi, Vahid, and Shomorony and resolves their conjecture. Our results also determine the capacity of the noisy shuffling channel of Shomorony and Heckel, in which all fragments have the same length, and resolve their open question.

We investigate a discrete analogue of the Fisher information. By using a discrete Laplacian operator, we can construct a discrete analogue of the score function and hence of the information. This provides a natural framework for establishing several information-theoretic inequalities in the discrete setting.

We study PAC learning of binary classifiers when labels arrive after random delays and are censored beyond an observation window $W$. How many samples does this observation model require? We propose a distribution-dependent answer, the Delay Complexity Dimension (DCD). The DCD combines the VC dimension with the exact label-information rate of the observation channel. Every learner needs $\Omega(\mathrm{DCD}/\varepsilon^2)$ samples. An oracle pseudo-label ERM matches this rate on the instance family of the lower bound, up to constants and logarithmic factors. On that family the DCD is within a constant factor of the classical censored-ERM complexity, and whether the DCD rate is achievable where the two separate remains open. The theory also yields a quantization scheme under which delay information lowers the residual label variance, an expedition policy with a $(1-1/e)$ guarantee for its version-space objective, and a phenomenon with no analogue in the no-delay setting. Under a fixed collection horizon, a moderate window can strictly beat immediate labels. Experiments on synthetic data and the Criteo conversion logs support these predictions. A budget-matched test of a companion labeler-diversity prediction is reported as a negative result.

In this paper, we investigate the impact of antenna beam pattern models on the beam placement problem in Low Earth Orbit (LEO) satellite networks. Unlike conventional beam placement studies that often rely on a fixed antenna pattern or a predefined half-power beamwidth, this work compares four representative gain models, namely the Bessel $J_1$ model, the combined Bessel $J_1+J_3$ model, the Gaussian model, and a 3GPP-TR 37.842 model. For each antenna pattern, the corresponding half-power angle is obtained and used to construct a user compatibility graph. The beam placement problem is then transformed into a Minimum Clique Cover problem and solved using a graph coloring-based algorithm. Numerical results compare the considered antenna models in terms of the number of active beams, load balancing, statistical channel gain-to-noise ratio (SCGNR), and signal-to-interference-plus-noise ratio (SINR). We analyze the impact of increasing the number of active beams on SINR-based spectral efficiency to explain the importance of carefully controlling the number of active beams. The results show that antenna beam pattern models affect both beam resource usage and link quality, highlighting the need for antenna-pattern-aware beam placement in LEO satellite networks.

Loss and noise limit the use of light to distribute entanglement, transmit quantum information, and establish secret keys. Gaussian states have a compact description in terms of means and covariance matrices, but their entanglement still requires global optimization. We develop a framework for computing Gaussian-restricted versions of the relative entropy of entanglement and Rains entanglement. An integral representation reveals a convex geometry under matrix geometric means and reduces Gaussian Rains entanglement to relative entropy minimized over Gaussian states with positive partial transpose. We obtain unique optimal reference states and additivity for independent Gaussian input states. Our convex-majorization algorithm has an explicit geometric convergence rate, uniform in the faithful regularization parameter and valid for pure input states. Controlled numerical integration and approximate convex subproblem solutions give a bit-complexity bound that is polynomial in accuracy bits for each fixed rational input covariance matrix, with explicit dependence on covariance size. These calculations bound extractable entanglement and, through the relative entropy of entanglement, extractable secret key. Teleportation simulation, energy allocation, and a convex optimization of passive-basis loss cores give bounds on quantum and private communication under an input-energy constraint, allowing arbitrary two-way classically assisted protocols. Two- and three-mode examples compare the bounds with achievable rates, including matching pure-loss benchmarks. Finally, we separate the Gaussian-restricted quantities from their unrestricted counterparts and give a photon-number truncation method with controlled error for computing the latter.

We develop and analyze an entropic descent method for a nonconvex optimization problem on probability simplices arising in classical and quantum secure communication. The optimization variables describe a randomized encoder, allowing its input distribution and stochastic preprocessing to be optimized jointly. An exact relative-entropy identity yields a channel-independent majorization bound and an explicit mirror-descent iteration with a fixed unit step size. We establish quantitative descent, convergence of objective values, and stationarity of accumulation points on their support faces for arbitrary finite classical channels, without assuming degradedness or less noisiness. For strictly positive transition matrices, a square-root parametrization and a Kurdyka–Łojasiewicz argument establish finite length of the transformed trajectory and convergence of the entire sequence, including limits on the simplex boundary. For less-noisy channels, a restricted convex formulation admits global convergence with an explicit \(O(1/t)\) objective-error bound. We extend the relative-entropy construction to noncommuting quantum outputs and derive monotone alternating updates for covariant encoders, including correlated preprocessing over a fixed block. Numerical experiments compare the method with analytic benchmarks and constrained optimization, reveal multiple attracting values, and demonstrate strict improvements from stochastic preprocessing.

We present an optimal tester for support size that determines whether an unknown probability distribution is supported on $n$ elements or $\varepsilon$-far away (in total variation distance) with sample complexity $\Theta(\frac{n}{\varepsilon\log(n)})$, improving the prior best $O(\frac{n}{\varepsilon\log(n)} \log(\min\{\frac{1}{\varepsilon}, n\}))$ due to Ferreira Pinto Jr. and Harms (STOC 2025). As a generalization, we provide a pretty good lower bound on support size.

Modern attack surfaces are inherently dynamic. Vulnerabilities emerge over time and they remain exposed to exploit until defenders discover and patch them. Artificial intelligence (AI) amplifies these dynamics by accelerating both offensive and defensive processes, making modern cybersecurity increasingly a competition among interacting time scales and resources. We capture this competition using a queue of branching processes that couples vulnerability discovery and resource-constrained patching with the evolving attacker population targeting each vulnerability. Our primary risk metric is the cumulative number of exploits a vulnerability experiences before it is patched. We derive the expected cumulative number of exploits under birth-death-immigration (BDI) and rise-decay (RD) models of attacker activity. Under a BDI model of attacker activity, we identify a phase transition: the expected cumulative number of exploits is finite only when attacker growth remains below both the vulnerability discovery rate and the effective patching rate. Across both attacker models, increasing the exploit rate does not necessarily increase the expected number of exploits because faster attacker departure can sufficiently limit the active attacker population. Finally, we show that equal proportional AI amplification of offensive and defensive capabilities need not preserve security: accelerating both sides can shift the defensive bottleneck and move the system into an unbounded-risk regime.

Madiman and Tetali established strong and weak fractional inequalities that subsume classical entropy subset inequalities such as Han's inequality and Shearer's lemma. In this work, we derive a decomposition of the gap in the strong fractional inequality and use it to establish the necessary and sufficient conditions for equality in the strong form of Han's inequality. As a consequence, we characterize the equality conditions for a strong Loomis-Whitnet-type projection inequality of Balister and Bollobas (2012). Finally, we apply the weak fractional inequality to an extremal graph theory problem, recovering and extending results of Sason (2022) and Boucheron et al. (2013), whose approaches rely on Han's inequality.

Solomonoff induction is a general solution for sequence prediction that started the field of algorithmic information theory around 1960. Let $M$ be the universal semimeasure. After observing a string $\omega_1 \cdots \omega_{n-1}$, it predicts the next bit $\omega_n$ to be $b$ with conditional probability proportional to $M(\omega_1 \cdots \omega_{n-1} b)$. Thus the error of this probability is \[ e_\omega(n) = 1 - \frac {M(\omega_1 \cdots \omega_{n-1}\omega_n)} {M(\omega_1 \cdots \omega_{n-1} 0) + M(\omega_{1} \cdots \omega_{n-1} 1)}. \] If $\omega$ a computable infinite sequence with a self-delimiting description of at most $k$ bits, then $\sum_n e_\omega(n) \le k\ln 2+O(1)$. Lattimore, Hutter and Gavane consider the problem of predicting computable patterns in a sequence, for example, the pattern of repeated bits $\omega = 00\,11\,00\,00\,11\,00\cdots$ They show that the error probabilities of the bits fixed by a computable pattern have limit zero, for example, for repeated bits, one has $\lim_n e_\omega(2n) = 0$. We show that for some computable pattern, the sum of $e_{\omega}(n)$ over positions $n$ from the pattern is not bounded. However, the sum of squares $(e_\omega(n))^2$ is bounded exponentially in the program length of the pattern. It is also shown that bits of $\omega$ that are fixed by a computable classification function $f:\{0,1\}^* \rightarrow \{0,1\}$, converges faster: the sum of $(e_\omega(n))^2$ is bounded by a polynomial in the program length for programs whose runtime is bounded by any fixed computable time bound.

Rate splitting multiple access (RSMA) has emerged as a promising non-orthogonal transmission mechanism to improve spectral- and energy efficiency, and user fairness in multi-antenna broadband channels. To further enhance its performance, in this paper we investigate a novel cooperative RSMA (C-RSMA) transmission framework assisted by reconfigurable intelligent surface (RIS) and user relaying. More specifically, we first formulate the common- and private-rate maximization problem satisfying the transmit power constraint, by jointly optimizing the transmit beamformers at the BS and the reflective beamformer at the RIS. The above formulated problem is an intractable and NP-hard problem as the optimization variables are coupled and the strong residual self-interference (SI) is involved. To conquer the challenge mentioned above, we propose a low-complexity alternative optimization approach to jointly optimize the transmit and reflective beamforming. Simulation results show that our approach is capable of enabling superior sum-rate performance than the RIS-assisted space division multiple access (SDMA), half-duplex RSMA, and SDMA without using RIS schemes.

The $\mathbb{Z}_{p^s}$-additive codes are subgroups of $\mathbb{Z}_{p^s}^n$. A $\mathbb{Z}_{p^s}$-linear generalized Hadamard (GH) code is a GH code over $\mathbb{Z}_p$ which is the Gray map image of a $\mathbb{Z}_{p^s}$-additive code. A recursive construction gives such codes $H_p^{t_1,\dots,t_s}$, of type $(n;t_1,\dots,t_s)$ and length $p^t$, and a partial classification of them is known. In this paper, we complete the classification of these codes for every prime $p$. The main tool is the derived code, obtained by puncturing the code outside a block of coordinates determined by its kernel. For a nonlinear code $H_p^{t_1,\dots,t_s}$ with $t_1\geq2$, the derived code is a $\mathbb{Z}_{p^{s-1}}$-linear GH code of the same family, although the mechanism for odd $p$ differs from the binary one. As a consequence, for $t\geq3$, there are $\mathcal P(t+1)-\mathcal P(t)$ nonequivalent such codes of length $p^t$ if $p$ is odd, and $\mathcal P(t+1)-\mathcal P(t)-1$ if $p=2$, where $\mathcal P(n)$ is the number of partitions of $n$. In particular, this completes the classification of the $\mathbb{Z}_{2^s}$-linear Hadamard codes. We also prove a conjecture on the codes with $t_1\geq2$ and $t_s=0$.

We study the $(K,N,S,M,K_c)$ distributed linearly separable computation (DLSC) setting, where a master employs $N$ workers with a per-worker computation load of $M$ to recover $K_c$ deterministic linear combinations of $K$ messages in the presence of up to $S$ stragglers. We propose a DLSC framework based on nested cyclic codes that achieves $M=\frac{K}{N}(S+K_c)$. For sufficiently large $K$, we establish the matching converse $M\geq\frac{K}{N}(S+K_c)$, proving the asymptotic computation load optimality of the proposed framework. By instantiating cyclic locally recoverable codes (LRCs) as base codes in the framework, the master can recover the desired $K_c$ linear combinations by contacting fewer than $N-S$ workers under many straggler scenarios. This essentially softens the straggler threshold without increasing the computation load or communication cost. Finally, we extend the construction to reduce the per-worker communication load by a factor of $m$, while achieving an asymptotically optimal computation load of $M=\frac{K}{N}(S+mK_c)$.

We study non-additive rank-metric codes contained in the space of symmetric matrices over a finite field. Our main focus is on symmetric $2$-codes. For odd $q$, we introduce a quotient construction for $\mathrm{Sym}_3(q)$ in which rank-one differences are described by isotropic vectors of a nondegenerate quadratic form on $\mathbb F_q^4$. This leads to a correspondence with partial ovoids of the elliptic quadric $Q^-(5,q)$. Using a construction of Cossidente and Pavese, we obtain a non-additive symmetric $2$-code in $\mathrm{Sym}_3(q)$ of cardinality $q^4+2q^3$, improving the previously known general lower bound. We also give an explicit matrix description of this family. Lastly, we present a block-based construction for symmetric $2$-codes that extends the three-dimensional improvement to all odd dimensions. In particular, for every odd $n\geq3$ and every prime power $q>2$, we obtain \begin{equation*} A_q^sym(n,2) \ge q^{(n^2-1)/2}+q^{(n^2-3)/2}+q^{(n^2-9)/2}, \end{equation*} which is strictly larger than the maximum cardinality of an additive symmetric $2$-code with the same parameters. This yields the first infinite family (as the matrix dimension $n$ varies) of non-additive symmetric $2$-codes whose size is larger than the largest achievable size of an additive code with the same parameters; before this, such a gap had only been established for $n=3$.