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arXiv2026

The Resurrection of Spectrum Spreading for 6G and Beyond: From Sinusoids to Chirps

Orthogonal frequency-division multiplexing (OFDM) and its sinusoidal subcarriers have underpinned the 4G and 5G eras, delivering high spectral efficiency and resilience to multipath fading through an efficient multicarrier architecture. However, as future systems move toward doubly dispersive environments driven by high-mobility applications and migration to mmWave/sub-THz bands, the time-invariance assumption underlying OFDM becomes increasingly difficult to maintain, and Doppler-induced degradation becomes prominent. While enhancements such as MIMO, advanced coding, and scheduling provide incremental remedies, they introduce additional overhead, because the sinusoidal subcarrier itself offers no inherent waveform-level robustness to Doppler impairments. Accordingly, two time-frequency spreading philosophies have emerged to improve Doppler resilience by distributing each symbol's energy across both dimensions of the time-frequency plane: (i) 2D isotropic spreading via the delay-Doppler (DD) domain, exemplified by the orthogonal time frequency space (OTFS) family, and (ii) sheared spreading via parameterizable chirps, exemplified by the affine frequency-division multiplexing (AFDM) family. In this article, we examine key considerations for future waveform design across these paradigms and argue that transitioning from the sinusoidal subcarriers of OFDM to the chirp-based subcarriers offers a viable design direction for improving Doppler robustness while retaining much of the mature OFDM infrastructure. This perspective also highlights the suitability of chirp-based waveforms for integrated sensing and communications (ISAC) and their extensibility to emerging physical-layer techniques. Overall, we argue that the transition from sinusoids to chirps is a technically motivated, compelling evolutionary direction for future wireless physical layer design.

eess.SP↗

arXiv2026

Near-Floor Geometry Is Generic: Leverage Dispersion in Trained Overcomplete Codes

An overcomplete code packs $F$ features into $d<F$ dimensions, so linear readout of one feature picks up cross-talk from the others, bounded below by the rank-trace floor $W(F,d)=(F-d)/(d(F-1))$. Proximity to this floor is read as evidence of an efficient arrangement. It is not: an i.i.d. code of the same shape attains it at every shape and load we measure (1.0000-1.0005 over 7 matched controls), and the reason is exact. For the gain-calibrated pseudoinverse the attainment ratio is $\frac{d}{F(F-d)}(\sum_i h_i^{-1}-F)$ with $\sum_i h_i=d$, so it equals one precisely when the leverage $h_i$ is equalised across features: distance from the floor IS leverage dispersion. On 40 released decoder matrices - 24 sparse-autoencoder decoders plus 16 MLP down-projections of a hybrid model - all sit above their matched control (1.0055-1.2018), and all reach guaranteed affine failure 4 to 40 sparsity levels earlier: where a random code of the same shape is still safe, a released dictionary already has a feature no affine rule recovers. Autoencoders trained by the same recipe on a randomly initialised copy of the same model return to the floor (1.0038-1.0065) in all 10 paired layers, so the excess reflects what the model learned, not the act of fitting an autoencoder. Under controlled conditions the loss decides whether an interface exists at all: $L^2$ reaches $R_{geom}=18.5635$ to 32.3131 with its frontier collapsed to $1$, against $1.0002$ for $L^4$, and that frontier follows a rate we registered before the widest run (10.79 predicted, 10.510 measured). Two limits are stated rather than implied: frontier statements are worst cases, not typical-case claims; and because the probe class contains the network's own decoder, our decoder comparison bounds supervised fitting rather than affine expressiveness.

cs.LG↗

arXiv2026

Rate-2/3 Girth-8 (3,18)-Regular Quantum LDPC Codes from Two-Branch Finite-Field Bases and CPM Lifts

We construct a rate-$2/3$ quantum low-density parity-check (LDPC) code from a $(3,18)$-regular two-branch finite-field base and a circulant-permutation-matrix (CPM) lift of degree $P=101$. The resulting code is a Calderbank--Shor--Steane (CSS) code with parameters $[[34542,23032,18]]$. Its distance is established by a computer-assisted proof: an explicit nontrivial logical operator gives the upper bound, and a complete symmetry-reduced enumeration excludes every nonzero vector of weight below 18 in both parity-check kernels. We also prove that every affine-in-$t$ CPM lift of the same base satisfying CSS orthogonality has distance at most 18 for every prime lift degree $P>19$. The construction has row weight 18 and column weight 3, and the Tanner graphs of $H_X$ and $H_Z$ separately have girth 8. Decoder experiments with log-likelihood-ratio (LLR) joint belief propagation (BP) and deterministic post-processing show no failures in $10^8$ trials at $p=0.01$, and a finite-length frame error rate (FER) sweep estimates the transition near $p=0.029$.

quant-ph↗

arXiv2026

Algorithmic List Decoding at Capacity and Optimal Proximity Gaps for Reed-Solomon Codes

We give a unified hidden-derivative framework for list decoding and mutual correlated agreement of ordinary Reed--Solomon codes over prime fields, on arbitrary prescribed evaluation sets. For every fixed slack $γ>0$, every sufficiently large block length $n$, every prime $q\ge n$, and every dimension $1\le k\le(1-γ)n$, a deterministic algorithm finds all codewords within relative distance $1-k/n-γ$ in $q^{O_γ(1)}$ time. The final list has size $n^{O_γ(1)}$, independently of $q$. Both statements extend to bounded-input-list recovery, with constants depending additionally on the input-list bound. For every fixed curve degree $\ell$, at most $n^{O_{γ,\ell}(1)}$ parameters on a curve $f_0+zf_1+\cdots+z^\ell f_\ell$ admit a nearby codeword whose exact agreement support is not a maximal jointly explained support of the coefficient words. For lines this gives MCA error $n^{O_γ(1)}/q$, with no proximity loss. The interpolation stage reparameterizes and optimizes the hidden-derivative construction of Brakensiek, Chen, Putterman, Zhang, and Zheng; differential root enumeration uses Kopparty's algorithm. We then prove that a specialization-safe differential equation has a cover by constant-dimensional varieties of polynomial cumulative degree, outside polynomially many parameter values. Intersecting these varieties with equations from the full agreement support yields both the field-size-independent list bound and exact-support MCA.

cs.IT↗

arXiv2026

Three Infinite Classes of APN Permutations on $Z_n$

For any permutation of a nontrivial finite abelian group, the differential uniformity is at least two; permutations attaining this bound are called almost perfect nonlinear (APN). We construct three infinite classes of APN permutations on the cyclic group $\mathbb{Z}_n$ using Singer cycles, binomials inducing projective permutations, and completed reciprocals combined with parity and quadratic characters. The respective domain orders are $q+1$ for prime powers $q>2$, $(3^d-1)/2$ for integers $d\ge2$, and $2p$ for primes $p>5$ with $p\equiv5\pmod6$. Each class contains an infinite subclass of composite orders outside the standard forms $r-1$, $r-2$, $r-3$, and $r-4$, where $r$ is a prime power. These forms arise in the Welch--Costas, Panario--Sakzad--Stevens--Wang, and Golomb constructions. To the best of our knowledge, these are the first infinite APN constructions on $\mathbb{Z}_n$ reported since 2011 that yield infinitely many composite orders outside these standard forms.

math.CO↗

arXiv2026

Cadence: Error-Bounded Lossy Compression of Demand Time Series with a Time-Series Foundation Model

We present Cadence, an error-bounded lossy compressor for numeric time series pairing a 330M-parameter time-series foundation model (Google TimesFM-3) with an adaptive arithmetic coder, guaranteeing $|\hat{x}_t-x_t|\leτ$ on every sample. One negative result constrains the design space: for lossless coding a foundation model is worth nothing, because bits saved are logarithmic in predictor accuracy, $Δb=\log_2(\mathrm{MAE_{old}}/\mathrm{MAE_{new}})$. So the $1.51\times$ advantage TimesFM-3 holds over a 32-tap linear predictor buys 0.60 bits of 20.28, a median gain of +0.03%. Error-bounded coding escapes this at one point: once a forecast lands inside the band the residual index is zero and the sample nearly free. Cadence contributes: (1) an adaptive range coder with context-modelled binarization, beating xz/zstd on real indices by 9.7% (15/15) and reversing a finding from a general-purpose back end; (2) a determinism result -- predictions are not bit-identical across batch sizes, and no PyTorch configuration repairs this, forcing group size and execution device into the container format; and (3) domain localization on corpora postdating any plausible training cutoff. On 49 EIA-930 balancing-authority demand series (2026) Cadence gains 13.3% over the best of six classical predictors, and 28.3% on 50 MTA ridership series (2026): 21.4% median over 297 series-tolerance pairs, winning all 297. Against downsampling, what time-series databases deploy for retention, its guaranteed worst-case error is $28$--$56\times$ tighter at equal size. End-to-end, once the context bootstrap is paid for, gains run from 6.8% at six months of hourly data to 15.1% asymptotically. Attempting to falsify the domain claim on SDRBench, theory predicts failure and delivers: -0.8% median, 0 of 27 pairs gaining. Three further negative results and eight retracted claims are reported in full.

cs.IT↗

arXiv2026

Fundamental Limits of Adaptive Beamforming Under Finite Training

Finite training reduces the output signal-to-interference-plus-noise ratio (SINR) of an adaptive beamformer, and a natural question is how much of this loss is unavoidable. This paper determines this question, providing a beamforming counterpart of the Cramér--Rao bound in spectral estimation. An exact identity expresses the SINR loss as a bounded function of the error in the clairvoyant minimum-variance distortionless-response (MVDR) weight. It yields a local asymptotic minimax lower bound over all measurable data-dependent beamforming rules, including biased and irregular rules. The first-order coefficient is $\tr(\Mb\Jb_{\rm eff}^{-1})$, where $\Jb_{\rm eff}$ describes the information in the training data and $\Mb$ measures the sensitivity of the output SINR. Matching constructions determine this coefficient in two complex-Gaussian models. For an $N$-sensor uniform linear array with $K$ distinct point interferers and $2K+1\le N$, a data-driven split one-step beamformer attains the coefficient $C_θ\le K$ at every interior scene of a fixed compact regular parameter set. For unrestricted covariance matrices, sample matrix inversion (SMI) attains the coefficient $N-1$ through the classical Reed--Mallett--Brennan law. The difference quantifies the first-order value of finite-source structure. Geometric formulas and numerical results describe the dependence on interference power and array geometry, and the finite-sample departure near a weak-source boundary.

cs.IT↗

arXiv2026

Low-rank matrix recovery landscapes beyond RIP with application to rank-one measurements

We study the problem of low-rank matrix recovery from linear measurements via the global nonconvex landscape of a low-rank factored formulation of the matrix LASSO (nuclear-norm--regularized least-squares). If the landscape is benign, that is, has no bad local optima, then practical and scalable algorithms can compute good statistical estimates. Previous state-of-the-art landscape guarantees have typically assumed that the linear measurement operator has the restricted isometry property, that is, the operator is approximately an isometry over all low-rank matrices. This is an unrealistic assumption for many applications; in particular, when the individual measurement matrices are themselves low-rank, we typically have poor upper isometry constants. To overcome this, we establish new guarantees of a benign landscape under a weaker isometry condition: rather than requiring upper isometry over all low-rank matrices, we only require it over the linear low-rank tangent space to the low-rank ground truth matrix. To illustrate the utility of this result, we apply it to the problem of matrix recovery from random rank-one linear measurements; via high-probability concentration bounds on the random measurement operator, we prove a novel landscape guarantee with statistically near-optimal sample complexity and recovery error.

math.OC↗

arXiv2026

Recovering linear images of sparse signals from indirect observations

In this paper, we develop and analyze techniques for recovering a linear image $Bx$ of an unknown signal $x$ from indirect noisy observation $ω=Ax+ξ$. It is {\em a priori} known that $x\in \cX$, a given convex compact set, and that $x$ is $s$-sparse---has at most $s$ nonvanishing entries. The proposed estimates belong to a large family of recovery routines by $\ell_1$-minimization. However, unlike the classical result describing performance of such estimates, we do not make any special (and hard to check) assumptions about the sensing matrix $A$ such as nullspace or Restricted Isometry condition and the like. As a consequence, parameters of the estimates and the upper bounds on their risks are not available in a closed analytic form, but are delivered instead by efficient computation as solutions to explicit convex optimization problems.

stat.ML↗

arXiv2026

Differentiable Horn Programs: A Constructive Expressivity Theorem for Latent Rule Operators

We introduce \textsc{LatentGamma}, a differentiable operator on the unit cube $[0,1]^M$ designed as a smooth surrogate of Tarski's immediate consequence operator $\TP$ associated with a definite Horn program $P$ on $M$ atoms. The operator is built as a five-stage composition combining sigmoidal gating, softmax routing, and a residual update that enforces monotonicity by construction. We establish four theoretical results. First, the iterated sequence is coordinate-wise non-decreasing and bounded by $\ind$, hence converges to a fixed point of \textsc{LatentGamma}; the operator itself is lattice-monotone on $[0,1]^M$. Second, our \emph{constructive expressivity theorem} shows that for every definite Horn program $P$ there exists a closed-form parameter assignment $θ^*(P)$ and an explicit time bound $T_{\max}(P)$ such that the iterated sequence \emph{exactly} reproduces $\TPinf(F_0)$ for every initial fact set $F_0$ throughout the window $[D(P, F_0), T_{\max}(P)]$, where $D(P, F_0) \leq M$ is the derivation depth. The window $T_{\max}$ is large in practice ($\geq 10^5$ for sparse programs) and reflects the finite-time nature of computation by smooth sigmoidal gates. Third, the oracle is robust to Gaussian noise on its body and head logits, with explicit non-asymptotic bounds. Fourth, we prove a matching information-theoretic lower bound on the parameter count. We provide a complete numerical validation on programs ranging from $M = 33$ to $M = 504$ atoms: oracle accuracy reaches $1.0000$ on $2000$ test cases with zero false-positive and false-negative rates, and the empirical noise tolerance $σ_{\max}$ scales precisely as the union bound $RM \cdot Φ(-10/σ_η)$ predicts.

cs.LO↗

arXiv2026

Rank and classification of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes

The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{α_1}\times\mathbb{Z}_4^{α_2}\times\mathbb{Z}_8^{α_3}$. A $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code is a Hadamard code which is the Gray map image of such a code. A recursive construction of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive Hadamard codes $\mathcal H^{t_1,t_2,t_3}$, with all $α_i\neq0$, $t_1\geq1$, $t_2\geq0$, and $t_3\geq1$, is known, as are the linearity and kernel of the corresponding codes $H^{t_1,t_2,t_3}$ of length $2^t$. Their rank, however, was not known in closed form. In this paper, we determine $rank(H^{t_1,t_2,t_3})$ for every admissible triple and obtain an explicit closed formula. The proof computes the dimension of the space spanned by the binary coordinate functions of the code, viewed as Boolean functions of the binary digits of the message. We then use the rank formula, together with the known kernel dimension, to classify the family as far as these two invariants allow. The only pairs of distinct types of the same length sharing both invariants are $\bigl(H^{1,2,t-6},H^{2,0,t-5}\bigr)$ for $t\geq7$ and $\bigl(H^{2,2,t-9},H^{3,0,t-8}\bigr)$ for $t\geq10$. Consequently, the number of distinct pairs $(r,k)$ among the codes of length $2^t$ is $\lfloor(t^2+6)/12\rfloor$ for $3\leq t\leq6$, $\lfloor(t^2+6)/12\rfloor-1$ for $7\leq t\leq9$, and $\lfloor(t^2+6)/12\rfloor-2$ for $t\geq10$. Thus, rank and kernel dimension classify the family completely if and only if $3\leq t\leq6$. We also determine all coincidences with the $\mathbb{Z}_4$-, $\mathbb{Z}_2\mathbb{Z}_4$-, and $\mathbb{Z}_8$-linear Hadamard codes of the same length: there is exactly one infinite family with the first two, namely $H^{2,1,t-7}$ and $H^{5,t-9}$ for $t\geq10$, and exactly five families with the $\mathbb{Z}_8$-linear codes.

cs.IT↗

arXiv2026

Scaling Bayesian Bandit Encoding with Shared Learning

A communication system must choose error protection and decoding effort as channel conditions change. A Bayesian bandit encoder (BBE) uses receiver feedback to learn which transmission configuration to select. We study a receiver that decodes by guessing error patterns, using Guessing Random Additive Noise Decoding (GRAND). We extend BBE's selection component to 1,008 code and decoder configurations by learning shared performance patterns offline and updating their weights online. Decoder noise models remain fixed. On a six-configuration training-selected shortlist, sharing reduces accumulated utility loss by 33.5% relative to independent learning. A fixed training-selected configuration matches the shared learner that searches the full catalog. After channel changes, the pruned shared learner first meets a near-optimal selection criterion in 88.5% of events by 2,000 packets, compared with 54.2% for pruned independent learning with the same discounting. The results support combining sharing and pruning for configuration selection, although packet losses remain high for the tested codes under severe noise.

cs.IT↗

arXiv2026

3D User Localization for Planar Arrays in LoS Near- and Far-Fields via Summed Phase Differences

This letter presents a phase-difference-based scheme for three-dimensional (3D) line-of-sight (LoS) user localization using a uniform planar array (UPA), applicable to both near-field and far-field regimes under the exact spherical-wave model. Unlike the previously studied two-dimensional (2D) uniform linear array (ULA) case, the 3D UPA case requires jointly exploiting the two array axes in order to recover the user's range, azimuth, and zenith angle. Adjacent-antenna phase-differences are first estimated from uplink pilots and then summed along the array axes to obtain unwrapped phase-differences between widely separated antenna elements. These summed phase-differences enable the construction of multiple three-equation systems whose solutions yield the user's range, azimuth, and zenith angle. We quantify the number of such equation systems, provide a representative closed-form estimator that uses only three phase-difference sums, and propose an all-data nonlinear least-squares estimator that exploits all available sums. Numerical results include a classical Fraunhofer near-field case and absolute 3D position RMSE in meters, and show near-CRB performance when the closed-form initialization is reliable. Moreover, unlike state-of-the-art baseline schemes, whose performance depends on well-tuned hyperparameters, the proposed estimators are hyperparameter-free.

eess.SP↗

arXiv2026

A New Approach to Code Smoothing Bounds

Code smoothing is a phenomenon in which an error distribution makes a code statistically close to the uniform distribution over the ambient space. This closeness is measured by the total variation distance. Recently, Debris-Alazard et al.\ introduced a smoothing bound, which is an upper bound on this total variation distance. Although the smoothing bound evaluates how the error distribution smooths a code, this bound applies only to linear codes. In this paper, we generalize this bound to not only linear codes but also specific non-linear codes. While the smoothing bound in previous work was obtained by Fourier analysis over finite abelian groups, we derive this bound using a graph-theoretic approach. To derive the smoothing bound, we consider code smoothing as the mixing of random walks on a specific graph, and use the concept of equitable partitions, which is well-studied in graph theory.

cs.IT↗

arXiv2026

Statistical Characterization and Block-EM Estimation of Frequency-Domain NSI for OFDM Systems in Bursty Impulsive Noise

Impulsive noise (IN), characterized by its high power and non-Gaussian distribution, poses a critical challenge in modern orthogonal frequency-division multiplexing (OFDM) systems, driven by the proliferation of electronic devices. Current IN mitigation techniques rely heavily on time-domain processing. These methods are applied before the discrete Fourier transform (DFT), introducing additional complexity, failing to align with OFDM's inherent frequency-domain processing flow, and risking the destruction of subcarrier orthogonality due to imperfect IN subtraction. To address these limitations, we propose a frequency-domain, block-based framework for mitigating IN. The statistical representation of IN in the frequency domain is first derived using a transformed Gaussian mixture model. Based on this model, we develop an optimal receiver that leverages perfect noise state information (NSI), thereby identifying scenarios in which NSI is critical. We then propose an unsupervised block-based expectation-maximization (EM) framework for NSI estimation and develop three variants for evaluation. These include a simple symbol-by-symbol variance-updated EM, a sequence-based transition-updated EM, and a MAP-based EM that exploits a sparsity-promoting prior to automatically prune the number of states. Our frequency-domain design operates after the DFT, seamlessly integrates with the OFDM processing chain, preserves subcarrier orthogonality, and leverages the known IN block structure to achieve substantial performance gains without the immense complexity of time-domain impulse reconstruction.

eess.SP↗

arXiv2026

Blind Random Search with Noisy Loss Measurements: Averaging, Thresholding, and Almost Sure Convergence

Blind random search repeatedly draws a candidate point and replaces the current estimate whenever the candidate has a lower loss. In the absence of noise, the true loss is observed directly. It decreases strictly at every accepted update and is monotone nonincreasing over all iterations. Measurement noise can make a worse candidate appear better and thereby break this monotonicity. To recover almost sure convergence under noise, we incorporate averaging and thresholding into the original decision criterion. These two classical tools are coupled. As the sample sizes grow, the positive threshold shrinks at a matched rate. These modifications allow blind random search to recover eventual monotonicity of the true loss under noisy measurements and to converge almost surely.

math.OC↗

arXiv2026

Ultra-High Resolution Method for Multipath Within a Co-Delay-Doppler Bin in DFT-P-OCDM

Communication systems can reuse their transmitted signals for sensing without dedicated radar transmissions. For an established DFT-preprocessed orthogonal chirp division multiplexing (DFT-P-OCDM) waveform, this task becomes difficult when several physical paths in a doubly selective channel fall into the same co-delay-Doppler bin. In this case, the number of resolvable delay classes inferred from the pilot may be smaller than the number of physical paths within the co-bin. This mismatch increases the difficulty of path number determination, fractional Doppler offset estimation, and channel reconstruction. We derive a pointwise relationship between the input and output in the DFT preprocessed Fresnel (DPF) domain for doubly selective channels with multiple paths within the co-delay-Doppler bin. Based on this input and output relation, we propose the two stage ultra high resolution (TSUR) framework. The first stage uses pilots of the phase progression to estimate delay, while the second stage uses the leakage samples to estimate the Doppler and the number of paths within each delay class. Furthermore, we derive CRLBs and analyze how the pilot configuration trades sensing resolution and communication recovery. Simulation results demonstrate that TSUR resolves same delay paths within a co-delay-Doppler bin, remains robust to exist delay offsets, and achieves lower fractional Doppler estimation errors than sequential extraction and off-grid baselines.

cs.IT↗

arXiv2026

Decoding Error Probability of the Random Code Ensemble over the Erasure Channel

In this paper, we provide explicit formulas for the average decoding error probabilities of the ensemble of all $(n,M)_q$ codes over the erasure channel under unambiguous, list and maximum-likelihood decoding principles. Moreover, we derive the asymptotic error exponents for the ensemble of all $(n,q^{Rn})_q$ codes.

cs.IT↗