Papers

Accounting for information flow on the path space of trajectories of a nonnegative martingale yields exact variational identities for it, even at arbitrary random times. This recovers the widely used classical concentration inequalities, from Ville to PAC-Bayes, and measures what each one discards. The tail a bound controls is itself a relative entropy, resolved by the chain rule into per-step conditional divergences. The discarded slack has an exact form in each of three geometries: a Gibbs tilt for the Azuma-Hoeffding and PAC-Bayes bounds, the crossing itself for Ville's and for pooled tests, and a dominating certificate for the $L^p$ maximal bound. That certificate's optional-stopping deficit resolves per step into Bregman divergences of the running maximum. On a path-time space, the same identity gains one factor that prices anticipation: an arbitrary random time carries an e-process “peeking penalty.” The partition function can be read as a coalescent–a prefix-sharing probability of independent copies–and geometric mixtures of test martingales gain a pooling benefit for multi-model safe testing.

We introduce the honeycomb hierarchy, a representation-theoretic framework that gives new asymptotic upper bounds on $R_2(\delta)$. Its first level is the two-row hyperoctahedral representation graph associated with type $S^{(n-k,k)}$. Retaining every two-row irreducible and every coordinate box-transfer channel, together with a moving-projection theorem, yields an explicit four-parameter exponent $\kappa_{\mathrm{HC}}$. The earlier whole-cube exponent $\kappa_H$ is a boundary restriction of this optimization, whereas the fully optimized second MRRW exponent $M_2$ is an exact symmetric slice. The prior best curve is the combined $\kappa_{\mathrm{bin}}=\min\{\kappa_{\mathrm{CW}},\kappa_H\}$, which uses a constant-weight branch $\kappa_{\mathrm{CW}}$. Replacing only the whole-cube branch by the honeycomb bound gives $\kappa_{\mathrm{best}}=\min\{\kappa_{\mathrm{CW}}, \kappa_{\mathrm{HC}}\}$. We prove, on $0<\delta<1/2$, \[ R_2(\delta)\le \kappa_{\mathrm{best}}(\delta) \le \kappa_{\mathrm{bin}}(\delta) \le R_{\mathrm{2MQC}}(\delta)<M_2(\delta),\\[-1mm] \kappa_{\mathrm{best}}(\delta) \le \min\{\kappa_{\mathrm{CW}}(\delta), \kappa_{\mathrm{bal}}(\delta)\} <R_{\mathrm{2MQC}}(\delta), \qquad \kappa_H(\delta)=R_{\mathrm{MQC}}(\delta). \] The hierarchy has two further directions. Increasing the representation depth replaces scalar by matrix-valued transfers on the hive. Increasing the anchor depth localizes it in a stable-set hierarchy. The resulting bounds are monotone in both directions and eventually recover $A_2(n,d)$. A complementary Horn–channel hierarchy gives matrix optimizations whose $2\times2$ level is $\kappa_{\mathrm{HC}}$ and whose $3\times3$ level is a stronger bound. Already at low levels, they can be used to improve the strongest previous general bounds, while the honeycomb framework provides a route towards tighter bounds.

Bernard and Letac (1971) introduced a method for uniform random sampling among $m$ outcomes from an unknown biased source of independent and identically distributed symbols. The process terminates when the multinomial coefficient of the cumulative symbol counts equals zero modulo $m$. This study extends the computational and information-theoretic analysis of their construction by presenting five algorithms with formal correctness guarantees and comprehensive complexity analyses. For prime $m=p$, the Bernard-Letac framework is analyzed in greater detail. The Rényi entropies of the source yield an exact product formula for the expected number of draws. A first-order approximation consistently overestimates this value, and the entropy lower bound is never attained. As $p$, treated as a continuous parameter, approaches 1, the expected cost converges to a constant greater than 1, determined by the entire source distribution. Furthermore, for every prime modulus $p$ and every finite alphabet $I$, an explicit automaton with $p + |I| + 3$ states computes the mod-$p$ first-passage kernel of the walk from the base-$p$ digits of its arguments, reducing the fair assignment cost from quadratic to nearly linear.

Random access is a central requirement in DNA-based storage systems: one would like to recover selected information symbols without sequencing the whole encoded object. A recent combinatorial model associates to a generator matrix $G\in F_q^{k\times n}$ the random variable $\tau_i(G)$, measuring the number of sampled columns needed to recover the information vector $e_i$. We study the cardinality-based extremal and finite-geometric aspects of simultaneous multi-symbol recovery. For a nonempty set $I\subseteq[k]$, let $\tau_I(G)$ denote the number of random column samples needed until all vectors $e_i$, $i\in I$, lie in the span of the observed columns. This variable interpolates between the singleton random access problem and the full-recovery problem underlying coverage depth. For each $m$, we introduce uniform worst-case and average parameters over all requested sets $I$ with $|I|=m$. Using the known subset-counting formula for $E[\tau_I(G)]$, we establish general upper and lower bounds for these parameters. In particular, the lower bounds are expressed through order statistics of the singleton recovery variables and specialize to the known singleton bounds when $m=1$. For systematic MDS encoders, we record an equivalent form of the known multi-symbol expectation formula and derive monotonicity and asymptotic consequences. For simplex encoders in arbitrary dimension, we obtain closed formulae in terms of Gaussian binomial coefficients; the full-recovery endpoint agrees with the known coverage-depth formula for simplex codes. Finally, in dimension three we study balanced quasi-arcs and compare their values with the simplex and MDS benchmarks.

Reliable multi-UAV communication control depends on predicting which aerial links will serve traffic before measurements are available. Radio world models (radio WMs) make such planning tractable, but their errors are nonuniform: a globally accurate model may still fail along high-demand corridors or association boundaries where rate errors reverse control decisions. This mismatch creates a learning challenge. Link queries must reduce decision-relevant channel uncertainty, while counterfactual trials must be filtered so that biased rollouts do not corrupt the policy. Existing acquisition and model-based control treat these budgets separately, valuing uncertainty, coverage, or optimistic return rather than risk reduction. We present RMWorld, a task-aware radio-WM framework that couples value-of-information channel calibration with credibility-diversity multi-trial selection. A biased propagation formula is corrected by a Bayesian residual, and each link is valued by its exact one-label reduction in locally linearized task-integrated posterior rate variance. Counterfactual branches are selected by a task-gated log-determinant objective, followed by conflict projection and fixed-batch validation. We derive the variance-reduction identity, prove posterior task-risk equivalence and the submodular greedy guarantee, and establish a scoped first-order non-interference result. Across 100 paired 3GPP trials RMWorld reaches 0.949~bit/s/Hz task-weighted RMSE, and across 30 severe-load DeepMIMO trials it reduces median backlog by 0.967 versus Ensemble UCB at 37.5% more offline rollouts.

With additional spatial degrees of freedom provided by the active intelligent reflecting surface (IRS), interference alignment (IA) can be achieved at low cost. In this letter, we propose a real IA scheme for an active IRS-aided system. The proposed scheme only requires the IRS to know the instantaneous channel coefficients under the assumption of blocked direct links. To maximize the achievable sum rate subject to individual minimum rate requirements and transmission power constraints, we propose a rate-profile learning-based algorithm. The algorithm uses offline-trained achievable rate profiles to decouple the original problem into multiple feasibility subproblems, which are then solved by generalized eigenvalue decomposition. Simulation results demonstrate that our proposed algorithm outperforms the conventional weighted minimum mean square error algorithm, while requiring significantly less program execution time.

We provide a positive answer to a conjecture proposed by Tohǎneanu and Van Tuyl regarding the minimum distance of codes whose underlying set of points is a reduced complete intersection. Despite the technical nature of the conjecture, we show that it follows directly from a not-well-known refinement of the classical Bézout bound for overdetermined polynomial systems. For completeness, this paper presents a self-contained proof of this refined bound. Furthermore, we show that using the same approach, it is possible to obtain a bound on the generalized Hamming weights of such a code and, more generally, to control the minimum distance of the codes obtained by evaluating forms of degree $d$ on the points of a zero-dimensional complete intersection.

Probability estimation over large alphabets under log loss is a well-studied problem, with celebrated methods such as the Good-Turing estimator. We introduce and study a new Bayesian estimator with four notable properties. First, its construction is exceptionally simple: multiply independent uniform draws from the probability simplex coordinate-wise and renormalize. Depth is the only structural parameter, and averaging over depths eliminates the need to tune it. Second, the regret of the resulting mixture, the excess code length it pays relative to a code that knows the source, admits an explicit and efficiently computable expression. Third, despite its simplicity and lack of tuned constants, the estimator is competitive across a diverse set of synthetic and real-text benchmarks with substantially more specialized methods, including Good-Turing. Fourth, the tractability of its regret allows us to identify scaling laws in data, alphabet size, and depth. For Zipf targets with exponent above one, the regret has a simple reading as long as the sample reveals only a small fraction of the alphabet. It closely matches the description length of the set of discovered symbols, at one bit of code per bit of description, plus a further cost per symbol. The data exponent is therefore the rate at which new symbols are discovered.

Let K_q(n,R) denote the minimum cardinality of a q-ary code of length n with covering radius R. We improve the known bounds on K_q(n,R) in 83 cases (82 distinct cells). On the upper-bound side we give 25 improved bounds for 5<=q<=15 – twenty-four found by search and one propagated by monotonicity – using two complementary methods: an engineered focused local search seeded with structural constructions, and a large-neighbourhood search driven by exact full-space coverage transforms that evaluates every candidate codeword position simultaneously. These are, to our knowledge, the first improvements to any upper bound on K_q(n,R) with q >= 5 since the 2011 revision of Keri's tables; several bounds decrease by more than 20%, e.g. K_6(8,4)<=166 (previously 216) and K_8(10,5)<=1883 (previously 2461). On the lower-bound side we give 58 improved bounds for 6<=q<=21, obtained from the semidefinite programming hierarchy of Gijswijt and Polak, whose published results cover q<=5, by combining an exact-arithmetic reimplementation of the reduced program with a multiprecision solution pipeline. Every new lower bound is certified by a rational dual solution validated by a standalone exact-arithmetic checker; no floating-point computation is part of the trusted base. The same pipeline also gives strong numerical evidence of limits: on a dozen further cells the certified value of the relaxation, which the solver reports as optimal to within its working precision, lies below the best known bound, indicating that no improvement is available there at this level of the hierarchy. One cell is improved from both sides: 441<=K_6(10,4)<=2751, previously 417–2952. All codes and certificates are provided in machine-readable form together with standalone verifiers.

Marton's inner bound, the best-known achievable region for a general discrete memoryless broadcast channel, was proposed by Katalin Marton in 1979, and whether it always achieves the capacity region has remained open since then. In this paper, we establish its strict sub-optimality: we show that the capacity region of some discrete memoryless broadcast channels can be strictly larger than Marton's inner bound.

Resilience has emerged as a fundamental capability for future wireless systems operating in dynamic and uncertain environments. Although resilience has attracted growing attention across academia, industry, and standardization, its conceptual scope, enabling mechanisms, and realization techniques remain fragmented. This paper presents a systematic framework of resilience in wireless systems from a trustworthiness perspective. We first formalize the concept of resilience and distinguish it from related uncertainty-aware terminologies, including reliability, robustness, adaptability, survivability, and recoverability. We then establish a hierarchical framework that organizes resilience into capability dimensions and enabling mechanisms, and that quantifies resilience through different technical aspects. We further use physical links and unmanned aerial vehicle networks as representative wireless scenarios to demonstrate how resilience can be systematically realized through the joint design of architectures, operations, and algorithms. Finally, we examine the fundamental trade-offs in resilience engineering: Improving resilience generally incurs costs in resource efficiency, nominal performance, information acquisition, implementation complexity, and latency.

Transmissive reconfigurable intelligent surfaces (T-RISs) integrated into the transmitter provide a viable realization of tri-hybrid multiple-input multiple-output (MIMO), where spatial processing is distributed across the digital, analog radio-frequency (RF), and electromagnetic (EM) domains. However, unlike conventional RIS-assisted links, a transmitter-native T-RIS directly participates in radiation formation, making T-RIS front-end modeling and weighted sum-rate (WSR)-oriented joint precoding challenging under practical hardware constraints. This paper develops a unified modeling and precoding framework for transmitter-native T-RIS tri-hybrid multi-user (MU) downlink transmission. Specifically, starting from a continuous-field description, the received field is characterized by the interaction among the feed array, the programmable T-RIS aperture, and the user-side propagation, leading to a cascaded baseband input-output model for MU precoding. Furthermore, the same representation is instantiated in the Fresnel, Fraunhofer, and mixed-field regimes, so that near-field focusing and far-field angular steering can be handled within one front-end model. Additionally, a WSR maximization problem is formulated over the digital precoder, analog network, and T-RIS coefficients under power and quantization constraints. A two-level solver is then developed by coupling outer weighted minimum mean-square error (WMMSE) updates with WMMSE-induced aperture-field shaping and hardware projection. Simulations validate the modeling accuracy and convergence, and show that the proposed full tri-hybrid design improves WSR over baselines while suppressing mixed-field cross-regime leakage.

A task-effective proposal is not yet a justified physical action. In networked Physical AI, a proposal may be understood while valid, timely, proposal-bound evidence or the authority required to finalize an action remains unavailable. We call this mismatch the verification gap and introduce a Post-Semantic Communication Framework for the systems interface between proposal formation and physical execution. The framework begins with application-declared evidence requirements, represents qualifying observations as evidence records, validates supporting and conflicting records through one path, and separates evidence sufficiency from authorized finalization and a downstream runtime gate. It further distinguishes evidence transfer, which can enlarge the record set reachable by a finalizer, from evidence coordination, which can suppress transmission around records already held at the finalization endpoint. Finite-state framework checks verify that the evaluator implements the declared distinctions consistently. Under the declared model, the controlled communication study exposes a finalizer-dependent asymmetry: sender-finalized Feedback uses evidence transfer to expand evidence reachability throughout the feasible plotted region, whereas receiver-finalized Feedback uses coordination to suppress redundant payload until loss, latency, freshness, and deadline costs shift selection to One-way. Finally, an episode-level reporting schema defines common denominators for future measured Physical-AI studies.

Weak arcs are point sets in PG$(n-1,q)$ meeting every general hyperplane (those are the hyperplanes not going through one of the points given by the standard basis vectors) in at most $n-1$ points. In this paper, we study weak arcs together with balanced variants which are contained on the sides of the fundamental simplex. We give an upper bound on the size of weak arcs, characterise the largest balanced quasi-arcs in the plane and construct large balanced quasi-arcs in PG$(3, q)$. We then use these configurations to build point sets for the random-access problem in DNA-based storage. The constructions are explicit, work over small fields, and attain recovery expectations matching the best known asymptotic bounds.

In safe hypothesis testing with test supermartingals, Ville's inequality provides anytime-valid type-I error guarantees for every significance level $\alpha\in(0,1]$, if one rejects the null hypothesis whenever the wealth process first exceeds $\frac{1}{\alpha}$. Due to an inherent asymmetry, the type-II error does not have such guarantees: a heavy concentration of the probability on the lower tail of the log-increments can lead to one catastrophic bet that undoes any amount of accumulated evidence. This paper studies how different hypotheses on those lower-tail probabilities lead to different bounds on the type-II error of the sequential test. They all reduce to one master inequality, which bounds the type-II error at level $\alpha$, at a fixed horizon and sequentially, in terms of a one-sided Legendre transform of the (inverse-)moment generating function of the e-variables, evaluated at one number: the amount by which the lower bound of the accumulated e-powers exceeds $\log\frac{1}{\alpha}$. And, the step is lossless, in the sense, that it extracts exactly a constrained information projection. Every bound presented here is a corollary, obtained by a certain majorant of the above function. The hypotheses are: a finite negative moment; an exponentially small crash probability with a moment on the winning side; a wealth floor with a conditional variance, and its Bernstein variant, which interpolates between a Gaussian regime set by the variance and an exponential one set by the scale; a sub-Gaussian or bounded-tilt lower tail; bounded log-increments; and i.i.d. increments, where the majorant is the truth. We also provide an empirical-Bernstein variant. Each hypothesis may either be read as a condition on the e-variables one has, or as the price of betting with an approximation to the likelihood ratio rather than the ratio itself, which satisfies the weakest condition for free.

In safe hypothesis testing with test supermartingales, Ville's inequality provides anytime-valid type-I error guarantees for every significance level $\alpha\in(0,1]$, if one rejects the null hypothesis whenever the wealth process first exceeds $1/\alpha$. Due to an inherent asymmetry, the type-II error behaves differently. We prove two things about the latter, for a simple null and alternative. First, the mean growth rate $\mathbb{E}_{P_1}[\log E]$, the e-power, that Kelly betting and growth-rate-optimal e-variables maximise, bounds nothing on its own. For every level $c>0$, every $\alpha$ and horizon $t$ we construct e-variables of conditional e-power exactly $c$ whose probability of not rejecting by $t$ is arbitrarily close to one. It forces eventual rejection, but no finite-horizon guarantee follows. Second, the quantity that does control the type-II error is the Chernoff-Stein exponent of an e-variable, $\Lambda(E)=\sup_{s\ge0}\{-\log \mathbb{E}_{P_1}[E^{-s}]\}$, whose range is exactly determined: $\sup_E \Lambda(E)=\mathrm{KL}(P_0\|P_1)$, the classical Chernoff-Stein exponent, and so the ceiling of its own per-e-variable form. One conditional application of Hoelder's inequality per step gives it, for every test supermartingale on an arbitrary filtered space, with no independence or product structure; the i.i.d. case adds that it is matched, and attained by nothing. The e-power has its own ceiling, $\mathrm{KL}(P_1\|P_0)$, and that one is attained, $P_0$-a.s. uniquely, by the likelihood ratio $R$. The two optima are the same divergence in opposite arguments, at opposite ends of the flattened family $R^{\beta}/\mathbb{E}_{P_0}[R^{\beta}]$: the ceiling as $\beta\downarrow0$, $R$ at $\beta=1$. Which $\beta$ is best is settled by the horizon, exactly: $R$ is optimal at $t=\log(1/\alpha)/\mathrm{KL}(P_1\|P_0)$ alone, beaten by sharpening $(\beta>1)$ below it and by flattening above.

A quantum $(r,\rho)$-locally recoverable code ($(r,\rho)$-qLRC) is a quantum code in which every qudit can be recovered from at most $r+\rho-1$ other qudits, even after $\rho-1$ additional erasures inside the recovery set. The bounds currently known for this class, namely the Singleton-like and the GG Singleton-like bounds, are alphabet independent and are therefore loose for small-to-moderate qudit dimensions. In this letter, we derive three alphabet-dependent upper bounds for pure $(r,\rho)$-qLRCs obtained through the Hermitian CSS construction: a Griesmer-like, a Plotkin-like, and a sphere-packing-like bound. We further establish the asymptotic hierarchy among these bounds and identify the relative-distance regions in which each of them yields the tightest rate constraint.

These notes develop four interconnected ways of reading a quantum circuit. A circuit for us begins as an operational composition of gates; then, it becomes a diagram whose local equalities may be used as calculations; next, it becomes a protected process once errors, syndromes, and logical degrees of freedom are separated; and finally, it becomes geometric when its connectivity, topology, and boundary data are treated as physical design parameters. The development begins at the level of bits and qubits before appealing to Deutsch's and Grover's algorithms as basic examples of quantum circuits. With the basics in hand, we interpret quantum circuits diagramatically, leading us to compact closed string diagrams and the ZX-calculus. After that, we consider how to correct quantum circuits by introducing the Knill–Laflamme condition, homological surface codes, and related concepts with a view towards thinking of these as operations on diagrams. The lectures eventually arrive at the properties of hyperbolic quantum codes and the prospect of physical superconducting circuits emulating the negatively-curved lattices needed to support those codes. These mathematical ideas and physical experiments, taken together, represent one way to impart a geometric layer onto quantum circuits. By the very end, we bring the ideas nearly full circle by assessing the extent to which these device physics experiments operationalize the basic ZX diagrams encountered much earlier in the story. While the later material reports on original research, and while the discussion becomes increasingly mathematical as the sections progress, no prior knowledge of quantum information, quantum computing, or quantum error correction is actually assumed.

Calderbank-Shor-Steane (CSS) codes, constructed from nested classical codes $C_2 \subseteq C_1$, are typically optimized for good code parameters. However, practical quantum computing equally demands fault-tolerant logical gates. In this work, we characterize nested pairs $(C_1, C_2)$ whose resulting CSS codes realize a target logical diagonal gate via transversal physical $Z$-rotations. In doing so, we recover a result of Camps-Moreno et al. that CSS codes can realize only logical single-qubit $Z$-rotations and multi-qubit controlled $Z$-rotations via transversal physical $Z$-rotations. Building on our characterization, we develop the ”appending construction”, that takes as input an $[[n',k']]$ CSS code $Q'$ and a target logical $Z$-rotation (single-qubit or multi-controlled) $U_L$, and extends $Q'$ by systematically appending $n''$ physical qubits to obtain an $[[n,k]]$ CSS code $Q$ with $n = n'+n''$ and $k=k'$. The target logical gate $U_L$ is realized in $Q$ by applying a well-chosen physical transversal $Z$-rotation to the $n''$ appended physical qubits. Moreover, any logical gate realized via transversal physical $Z$-rotations in the input code $Q'$ remains transversally realizable in the extended code $Q$. The CSS code $Q$ may incur a loss in minimum distance, but the loss can be controlled through the parameter choices made in the construction. By repeatedly applying the appending construction, we can extend any CSS code $Q'$ to obtain a CSS code $Q$ that supports fault-tolerant implementations of multiple desired logical $Z$-rotations. The cost to be paid for this is the increased physical qubit overhead as the number of target logical gates grows. To illustrate our methodology, we construct CSS code families that transversally realize addressable logical single-qubit and multi-controlled $Z$-rotations.

Among $n+1$ equiprobable equal-energy signals in $\R^n$ under additive white Gaussian noise with maximum-likelihood decoding, which arrangement maximizes the probability of correct decoding? The question is Shannon's, recorded by Rice in 1950. Mulgund proved in 2026 that the regular-simplex value bounds the correct-decoding probability of every signal set at every signal-to-noise ratio, leaving open whether the simplex is the only maximizer. This paper determines the equality cases in a form stronger than uniqueness. A signal set other than a regular simplex falls strictly below the bound at every positive signal-to-noise ratio. Hence a code meeting the bound at one positive operating point is already a regular simplex, up to vertex relabeling and an orthogonal map. In probabilistic form, among the correlation matrices that signal sets induce, any matrix other than the identity gives a lower-orthant probability strictly above its independent counterpart at every finite threshold, leaving no room for a nontrivial equality. No code of ambient dimension below $n$ attains the bound. Under an energy budget $E$ with unrestricted blocklength the optimal codebook is uniquely the regular simplex of circumradius $\sqrt{E}$. Every optimal codeword therefore exhausts its allowance. Equality in the Simplex Mean Width Conjecture likewise occurs only at the regular simplex. The proof strengthens the first self-convolution step of Mulgund's argument with Royen's correlation theorem. The single-parameter rigidity is machine-checked in Lean 4.

Additive codes over finite fields generalize linear codes, and additive MDS codes provide a natural extension of linear MDS codes. In this article, we study additive twisted Reed–Solomon (TRS) codes and obtain new constructions of additive MDS codes. First, for additive TRS codes with twist $t=2$ and an arbitrary hook, we establish necessary and sufficient conditions for the codes to be additive MDS, thereby generalizing the results in Section 3 of [Jiayu Ma et al., New families of additive non-Reed-Solomon MDS codes]. In particular, we show that the existence of an additive MDS TRS code with $t=2$ and hook $h=0$ yields codes of larger lengths than those obtained for $t=2$ and $h=k-1$ in [Jiayu Ma et al., New families of additive non-Reed-Solomon MDS codes]. Next, we consider additive TRS codes with twist vector $\mathbf{t}=(1,2)$ and hook vector $\mathbf{h}=(0,0)$, and derive necessary and sufficient conditions for them to be additive MDS. We further establish the existence of such codes. Using the Schur square technique, we obtain mild conditions under which the constructed families are inequivalent to additive Reed–Solomon (RS) codes. Finally, we determine parity-check matrices for both families of additive MDS codes considered in this article.

This paper studies integrated sensing and communications (ISAC) over a hybrid system that seamlessly combines legacy cellular base stations with distributed cell-free (CF) access points (APs). We propose a hierarchical ISAC architecture where a central base station (CBS) serves its near users and simultaneously operates as a monostatic radar for aerial target detection, while distributed APs—many idle under user-centric clustering—act as cost-free bistatic receivers. The CBS jointly handles communication processing and multi-static sensing fusion, reducing fronthaul overhead compared to conventional cell-free ISAC. To achieve this, a five-phase time-division duplexing workflow with precise ISAC role assignment is specified. Closed-form expressions for spectral efficiency and multi-static sensing signal-to-noise ratio analytically characterize the communications–sensing Pareto frontier. Numerical results confirm that the proposed hierarchical design simultaneously achieves higher sum throughput and superior sensing accuracy than conventional cell-free ISAC.

We consider a fundamental problem of mixedness testing: Given $n$ copies of an $N$-qubit state $\rho$, determine whether $\rho = \mathbb{I}_d/d$ or $\|\rho-\mathbb{I}_d/d\|_1 \geq \varepsilon$ with high probability, where $d = 2^N$. In particular, we focus on performing this task in the practical setting of single-qubit measurements, where measurements are prepared independently on each qubit. We provide a nearly complete picture of single-qubit mixedness tesing by showing $n = \tilde{\Theta}\left(\sqrt{10}^N/\varepsilon^2\right)$. To establish our lower bound, we introduce a measurement-dependent lower bound framework for adaptive single-copy state certification. For the upper bound, we present a randomized Pauli basis measurement protocol, which relies on a new primitive for computationally efficient uniformity testing of correlation-concentrated distributions on the Boolean hypercube. In conjunction, we provide lower and upper bounds for mixedness testing with fixed Pauli measurement protocols.

We study radio node (RN) placement for indoor enterprise networks. Using stochastic geometry (SG), we derive the meta-distribution (MD) of the SINR for a test user equipment (UE), with and without cooperation from outdoor macro base stations (MBSs), and compare these results with an integer linear programming (ILP) approach. SG provides an estimate of the required number of RNs but not their locations, while ILP can yield inaccurate local optima and requires high computational power. To address this, we investigate clustering-based algorithms for initializing RN locations using UE location distributions. Along with standard methods, we propose a weighted $k$-harmonic means (WKHM) clustering strategy tailored to maximize SINR. We then introduce a constrained sequential minimum cut algorithm, SeqMinCut, to merge multiple RNs into larger cells and further improve SINR. This is the first work that integrates SG-based statistical analysis, optimization, and clustering to obtain system design insights, dimensioning rules, and planning strategies for enterprise 5G.

Stacked intelligent metasurfaces (SIM) provide an efficient architecture for integrated sensing and communication (ISAC) with few radio-frequency (RF) chains. However, diagonal SIM provide only element-wise phase control, so balancing multiuser communication and sensing performance may require additional layers. In this letter, we propose a beyond-diagonal SIM (BD-SIM) architecture for ISAC, enabling controllable intra-layer coupling through reconfigurable impedance networks, thereby enhancing wave-domain processing flexibility. We develop a unified alternating optimization framework applicable to fully-connected, group-connected, and diagonal SIM architectures. Within this framework, we derive a closed-form power allocation rule and propose an effective variable separation algorithm for multi-layer phase-shift design. Simulation results show that the proposed BD-SIM achieve a better communication-sensing trade-off and require fewer layers to attain performance comparable to conventional SIM.

This paper investigates the construction of linear codes via abelian codes over finite fields. By exploiting the algebraic structure of multivariate polynomial quotient rings, we derive lower bounds on the minimum distance using a generalized shift bound, which extends the classical van Lint-Wilson bound for cyclic codes. Several infinite families of abelian codes are explicitly constructed, including binary and ternary cases that extend previously known cyclic constructions. To find more abelian codes with good parameters, we apply a genetic algorithm that searches over defining sets represented as binary chromosomes of cyclotomic cosets; the fitness function compares the computed minimum distance against the best known linear code (BKLC) bounds. The search yields multiple record-breaking codes over F_3 and F_4, with improvements over Grassl's tables. Furthermore, the nested structure of these codes enables the application of Construction X, yielding additional linear codes with improved parameters. The results demonstrate that abelian codes, combined with heuristic search, form a viable way for discovering linear codes with unknown parameters.

Encoding quantum information with low circuit overhead is a fundamental challenge in fault-tolerant quantum computation. Random circuits provide a natural mechanism for rapidly spreading logical information through simple gates applied in parallel. Brown and Fawzi showed that random Clifford circuits on two-qubit Clifford gates provide such encoders that achieve the quantum Gilbert-Varshamov rate-distance tradeoff with depth $O(\log^3 n)$. We show that the same asymptotic tradeoff is attained in optimal $O(\log n)$ depth under a gate distribution with a more restricted support. For every fixed $\delta>0$ and sufficiently large $n$, if $\frac kn < 1 - H(\frac{d}{n}) - \frac{d}{n}\log_2 3 - \delta$, we can construct random circuits of depth $O(\log n)$ which define, with high probability, an $[n,k]$ stabilizer code of distance at least $d+1$, which matches the $\Omega(\log n)$ light-cone lower bound for linear distance encoders. Our ensemble employs a random matching circuit architecture consisting of $T$ independent permutation-invariant layers. In each layer, the qubits are paired up by a uniformly random perfect matching, and a random independent two-qubit Clifford gate is applied to each pair. The gate distribution need not be uniform over, or even have full support on, the two-qubit Clifford group; rather, we allow for very general distributions on Clifford gates satisfying three regularity conditions. In particular, the construction can be implemented using $n/2$ CNOT gates on randomly matched pairs in each layer, with parallel one-qubit Clifford twirls. These regularity conditions allow us to reduce the second-moment dynamics of our random circuits to a reversible Markov chain on binary support strings. We establish logarithmic hitting-time bounds for this Markov chain and comparisons of its stationary distribution to prove the coding properties of the circuits.

Large antenna arrays allow wireless systems to serve more users and achieve higher data rates, but they also make channel feedback expensive: the receiving device must repeatedly report a large complex-valued channel matrix to the base station. Most neural compressors treat this matrix like an image and replace it with a fixed-length code that only a matched neural decoder can interpret. The message therefore does not adapt to channel complexity, and changing the antenna count typically requires retraining. We ask whether a device can instead report only the few dominant propagation paths underlying each channel. We introduce the Gramian Chebyshev Neural Operator (GCNO), a physics-based, variable-rate compressor that identifies a sample-dependent set of path directions. GCNO uses receive-transmit channel structure to locate paths, a first-order Taylor correction to refine directions that fall between grid points, and least squares to recover their complex strengths. It is trained without path labels, and the base station reconstructs the channel analytically from the transmitted path tuples rather than through a learned decoder. Across three ray-traced environments, GCNO achieves better reconstruction accuracy at the same payload - or lower payload at the same accuracy - than neural feedback baselines, and transfers to unseen antenna counts without retraining.

This tutorial introduces the algebraic foundations underlying the weight structure of polar codes. Using a monomial-based polynomial formalism, we explain how polar and Reed-Muller codes can be viewed as decreasing monomial codes, enabling systematic characterization and enumeration of low-weight codewords. Its goal is to provide an accessible introduction to affine automorphisms, orbit-based descriptions of minimum and low-weight codewords, and their role in weight enumeration. Through illustrative examples and high-level overviews of recursive and coset-based techniques, the paper aims to prepare readers for deeper engagement with the recent technical literature on the weight distribution of polar codes.

This paper investigates the sensing-centric design of reconfigurable intelligent surface (RIS)-enabled rate-splitting multiple access-integrated sensing and communication (RSMA-ISAC) systems. Specifically, we propose a new beam-gain approximation method to enhance the sensing beam gain while satisfying communication quality-of-service (QoS) constraints.Since the joint optimization of the beamforming vectors and RIS phase shifts is highly coupled and non-convex, existing methods typically rely on generic optimization solvers involving substantial computational complexity. To address this issue, we propose an efficient constraints-separation-based alternating optimization algorithm (CS-AO). Our proposed algorithm effectively decouples the optimization variables and yields closed-form solutions for all subproblems, thereby significantly reducing the computational burden. Simulation results show that the proposed algorithm achieves sensing beam-gain performance comparable to successive convex approximation (SCA) and semidefinite relaxation (SDR) benchmarks, while achieving more than 120-fold and 50-fold runtime reductions. In addition, compared with conventional space-division multiple access (SDMA) schemes, the proposed design exhibits substantial sensing beam gain.