communications engineers

Abstract: We develop design methodologies for the rate profile of polarization-adjusted convolutional (PAC) codes tailored to successive cancellation list (SCL) decoding. The crux of our approach is two-fold: first is the creation of a modified density evolution Gaussian approximation (DEGA), called conditional DEGA (CDEGA), to analyze the decoding process in the presence of errors; second is the creation of a dangerous-path occupancy metric to find the conditional occupancy of bit-errors in a finite list. Our approach not only illuminates the role of rate-one precoding, but also provides insight into why certain information sets are particularly well suited to PAC codes. We demonstrate the usefulness of CDEGA and dangerous-path metrics with algorithms that create new information sets, as well as precoders that are extremely sparse yet competitive. In particular, we design codes of length $N \leq 512$ with various rates and list sizes, all of which offer at least half a dB gain over traditional PAC codes at an FER of $10^{-3}$ and improve upon state-of-the-art results.

Abstract: This paper studies when list-decodability guarantees for error-correcting codes of good distance can go beyond the classical Johnson radius and gives a corresponding efficient list-decoding algorithm for Reed-Solomon codes up to this new radius. The Johnson bound shows that any error-correcting code with distance $δ$ is automatically list-decodable with small list size up to a certain radius $J(δ)$ around any received word $r$. We show that if the received word $r$ is known to have some codewords that are somewhat close to it (we call these ``hints"), then $C$ is list-decodable with small list size up to an even larger radius around $r$. We determine the optimal list-decoding radius in this setting for single and multiple hints over both large and small alphabets and show that this radius is the optimum of a natural convex program. Remarkably, even a single hint that is $(J(δ)-ε)$-close to $r$ improves the list-decoding radius around $r$ to $J(δ)+ε'$. Finally, we give an efficient algorithm, based on the weighted Guruswami-Sudan decoder, that list-decodes Reed-Solomon codes up to this optimal radius.

Abstract: We quantify the error of the strict-radius Kraskov--Stögbauer--Grassberger (KSG-1) mutual-information estimator on a Gaussian channel as the sample size \(n\) grows and the channel becomes nearly deterministic. For fixed dimension \(p\) and neighbor index \(k\), we define the resolution parameter \(u_n=ne^{-I_n}\), where \(I_n\) is the true mutual information in nats. When \(u_n\to u\in(0,\infty)\), we prove that the estimation error converges in mean square to a computable deterministic bias \(B_{p,k}(u)\). Sampling fluctuations therefore vanish, while an additive bias can persist. We derive this bias from an explicit marked-Poisson law for the neighborhood counts, characterize its small-\(u\) asymptotics, and show that it tends to zero as \(u\to\infty\). The same law describes how often neighborhoods saturate, meaning that they make the largest possible contribution to the estimate. The limiting saturation probability decreases continuously and strictly from one to zero, allowing a Gaussian-model bias correction on fixed critical intervals. Finite-sample experiments show substantial bias even when the true information lies below the estimator's logarithmic ceiling and saturated neighborhoods are rare. Saturation-based correction reduces estimation error in these experiments, though recovering the resolution parameter from the observed saturation fraction can remain difficult when saturation is rare.

Abstract: In this paper, we investigate a family of trace codes over finite fields of odd characteristic defined by two quadratic trace conditions. By analyzing the quadratic forms associated with these trace conditions and evaluating the corresponding character sums, we obtain the parameters and complete weight distributions of the codes. We further exploit the natural multiplicative orbit structure of the defining sets to obtain punctured codes and determine their parameters, weight distributions and dual distances. Moreover, we establish self?orthogonality criteria and investigate divisibility and projectivity of the constructed codes. These results give a unified treatment for arbitrary odd prime fields and for extension degrees of both parities.

Abstract: The canonical $\mathbb{Z}_{2^s}$-linear Hadamard codes are the Gray images of the $\mathbb{Z}_{2^s}$-additive codes. Their rank, the dimension of the binary linear span, was known in closed form only for $s\le 3$, and for $s \ge 4$ only through computer computations for small lengths. We give a method that computes the rank for every $s$. We show that the rank is a binomial polynomial in the type parameters, with coefficients $ρ_s(\boldsymbol{k})$ independent of the length. We compute all the coefficients for $s\le 5$, which yields explicit rank formulas over $\mathbb{Z}_4$ and $\mathbb{Z}_8$ (recovering the known ones), and over $\mathbb{Z}_{16}$ and $\mathbb{Z}_{32}$.

Abstract: In this paper, we focus on decoding MDS linear block codes for insdel errors. Building on our recent work for two-dimensional Reed-Solomon codes, we present a decoding algorithm for bounded distance decoding for the general insertion-and-deletion scenario that can be used for any MDS code of rate at most half. Our algorithm employs a sliding window and has a time complexity smaller than brute-force decoding. In particular, for optimal Reed-Solomon codes, it corrects insdel errors up to the half-Singleton bound, which exceeds the scope of existing methods. Additionally, for the special cases of insertion-only and deletion-only insdel error patterns, we present two different algorithms with a time complexity smaller than our general algorithm.

Abstract: We consider the problem of binary sequence reconstruction where a sequence is transmitted multiple times over a channel, and each undergoes substitution errors by an arbitrary error vector of Hamming weight exactly $t$. We determine $N_{\mathrm{wc}}$, the minimum number of distinct channel output observations guaranteeing unique reconstruction by an optimal decoder, and show that the bit-wise majority-vote decoder, which was proved by Levenshtein (2001) to be optimal for channels corrupting up to $t$ bits, is in fact suboptimal here. Despite this suboptimality, we prove a condition that admits a practical early-termination rule, necessary for every decoder at any number of outputs $N$, and sufficient for bit-wise majority-vote even below $N_{\mathrm{wc}}$. We evaluate the number of outputs at which this condition is triggered, numerically, and show that it frequently occurs below $N_{\mathrm{wc}}$ when the block-length $k>2t+1$. Finally, we propose an alternative algorithm aimed at guaranteeing unique reconstruction with fewer observations than bit-wise majority-vote, and demonstrate through exhaustive enumeration that it successfully decodes channel output-sets where bit-wise majority-vote fails.

Abstract: In this paper, we investigate the Linear Complementary Pairs (LCPs) of Maximum Distance Separable (MDS) codes that arise from elliptic curves. We establish geometric conditions for LCPs of elliptic codes and prove that they cannot consist entirely of MDS codes when evaluated over all rational points of the curve. To address this, we construct LCPs of MDS elliptic codes by restricting evaluation divisors to subgroup cosets. We apply this method to obtain explicit infinite families of LCPs of MDS codes over Mordell and supersingular elliptic curves, including large-length constructions over quadratic extensions.

Abstract: For every prime power $q>19683$, we settle the Cheng--Murray deep-hole problem for full-length Reed--Solomon codes in every dimension $2\le k\le q-2$, with the characteristic-two cases stated explicitly. For odd $q$, the deep holes are precisely the words of degree $k$. For even $q$ and $k\ge3$, the same conclusion holds except for an additional family when $k=q-3$. In even characteristic and dimension two, every deep hole has a unique expression $a h(X)+bX+c$, where $a\ne0$ and $h$ is a normalized $o$-polynomial. The proof combines quadratic quotient algebras, graph-theoretic weighted independent-set inequalities, and planar arc completion. We also obtain normal rational curve rigidity in the corresponding nonexceptional dimensions, the projective Reed--Solomon covering radii, and distinct-coordinate solvability and remainder criteria.

Abstract: Two-dimensional (2D) convolutional codes are a generalization of one-dimensional (1D) convolutional codes and are naturally suitable for encoding two-dimensional data. Although their theoretical properties are well established, practical applications are limited due to the lack of error-correction decoding algorithms. While the Viterbi algorithm provides an efficient decoding method for 1D convolutional codes, no analogous approach exists for the 2D case. In this paper, we introduce a decoding algorithm for 2D convolutional codes, called the 2D Viterbi algorithm, extending the principles of the classical Viterbi algorithm to the two-dimensional setting. The presented 2D Viterbi algorithm works with linear systems representations of 2D convolutional codes and its complexity depends on the dimension of the corresponding 2D linear system. Hence, we also provide several algorithms to obtain 2D realizations of small dimension.

Abstract: In the seminal paper [13], Farrán and Munuera introduced a method to establish generic information on the $m$th Feng-Rao distance via a function on arbitrary numerical semigroups that serves as a lower bound on the generalized Hamming weights of dual one-point algebraic geometry codes whenever the semigroup is Weierstrass. In the present paper, we study what we shall call the primary and dual, respectively, $m$th relative Feng-Rao distances, which serve as lower bounds on relative generalized Hamming weights of nested primary and dual, respectively, one-point codes under similar condition. A unified method for establishing generic formulas for $m$th relative Feng-Rao distances of general numerical semigroups is developed and for a large variety of numerical semigroups generated by two elements, closed formula expressions are established complementing Delgado et al.s work [8] on the $m$th Feng-Rao distances of such semigroups. Finally, for non-singular $C_{ab}$ curves, we develop a method to estimate the $m$th relative generalized Hamming weight of dual and primary one-point codes in those regions where the generic methods may fail to provide best possible results.

Abstract: In this paper, we determine the generalized Hamming weight hierarchies of two classes of twisted Reed--Solomon (TRS) codes. For TRS codes with one twist, we first determine all possible values of the generalized Hamming weights. We then give a complete characterization of the weight hierarchy in terms of the minimum distance of the dual code. Explicit conditions are derived for the minimum distances of the code and its dual, which make the whole weight hierarchy explicitly computable. As special cases, the obtained results recover several known characterizations for TRS codes to be MDS, NMDS codes, and $s$-gap AMDS codes. For TRS codes with two twists, we similarly determine the complete generalized Hamming weight hierarchy. In this case, the higher generalized Hamming weights are determined by the first two generalized Hamming weights of the dual code, for which explicit rank conditions are also obtained. In addition, we further determine the subcode support weight distributions of this class of single-twist TRS codes. More precisely, for each $1 \le r \le k$, we derive an explicit formula for the number of $r$-dimensional subcodes with a prescribed support size, thereby giving a complete description of the support distribution of subcodes.

Abstract: In this paper, we present a method for constructing linear codes derived from book graphs. Applying such a method, we construct binary codes that have the greatest possible minimum distance for their length and dimension by means of the $2$-sum of two equal book graphs. We then utilize matroid and graph theory together to investigate some properties of the resulting linear codes. Furthermore, we show that, from the incidence matrix of any graph, we can obtain a linear code over any finite field maintaining the same parameters. Furthermore, we show how the condition for two linear codes to generate a Calderbank-Shor-Steane (CSS) code translates into matroid theory and, utilizing cycle graphs, we exhibit a way to generate codes that meet this condition.

Abstract: We present a version of the vector balancing problem in which each vector may be given a sign and a permutation of its coordinates. We prove that this vector balancing problem and its corresponding prefix problem admit an explicit bound, and we further show that it is asymptotically optimal in the dimension. Our method of proof is purely geometric.

Abstract: The security of keyed modulation is often argued from the key-space size and the error rate of a key-less receiver. This evidence fails when the key is reused and the waveform is harmonically coupled. For a phase-keyed Fourier-curve constellation, whose $k$ tones share one data parameter, integer relations among the harmonic indices yield data-cancelling mixed moments of the received tones that expose key characters. A modular relation lattice characterizes the exposed characters; for consecutive harmonics, third-order moments recover the relative phases and a fourth-order moment completes the key up to cyclic relabeling whenever its coefficient is nonzero, as in all evaluated settings. A non-data-aided relation-moment estimator turns this leakage into an attack that never enumerates the key space. On a regular $(3,6)$ LDPC-coded link, one key per 168-symbol codeword leaves the eavesdropper a block error rate below $0.04$ at the middle noise level, and the attack meets a predeclared $0.1$ compromise criterion in eleven of twelve operating points. Tangent artificial noise and a harmonic set without relations below order four raise her measured error rate at intermediate reuse lengths but do not remove the one-codeword vulnerability. For a grid of $2^{128}$ protocol keys at the middle noise level, equal-length refresh schedules that keep a $95\%$ lower confidence bound of her block error rate above $0.9$ consume at least $1.52$ fresh key bits per information bit, $1.52$ times the entropy rate of a one-time pad on the data.

Abstract: We introduce a new family of recursively constructed sparse matrices, termed Fractus matrices, and investigate their use in constructing low-density parity-check (LDPC) codes. Generated through a self-similar recursive process, these matrices yield regular sparse parity-check matrices while preserving key structural properties across successive iterations. This recursive structure enables an efficient encoding algorithm with computational complexity that is nearly linear in the block length. Decoding is performed using standard iterative message-passing algorithms, thereby retaining the low-complexity decoding characteristic of LDPC codes. The proposed construction produces Tanner graphs with girth six and guarantees a minimum Hamming distance of at least $\ell+1$. We establish several algebraic properties of Fractus matrices, including sparsity, regularity, recursive decomposition, and symmetry under the flip-transpose operation. In addition, we show that the family of Fractus matrices admits a natural lattice structure and that the associated LDPC codes inherit corresponding lattice-theoretic properties. These results establish a connection between order theory and coding theory. Overall, the proposed framework integrates recursive matrix constructions, efficient encoding, graph-theoretic analysis, and lattice theory into a unified algebraic approach to the design and analysis of scalable LDPC codes.

Abstract: Error-correcting codes (ECCs) are essential across diverse applications, from wireless communications and storage to quantum computing, yet each application imposes distinct design requirements on the parity-check matrix (PCM). To address these on-demand requirements in a unified framework, we propose ZeroCode, a reinforcement learning (RL)-based approach that constructs PCMs sequentially from the all-zero matrix. ZeroCode formulates construction as a discrete sequential decision-making problem and uses proximal policy optimization with action masking to select valid edges. ZeroCode achieves a gain of approximately 1 dB over the prior RL-based construction method at a bit error rate (BER) of $10^{-4}$ for the (32,16) code and outperforms existing genetic, differentiable, and classical code-design methods in our experiments. Beyond optimizing decoding performance, the masking mechanism allows on-demand structural constraints, such as a maximum degree, 4-cycle-free structure, and quasi-cyclic structure, to be flexibly incorporated. Moreover, a single policy rollout yields a library of PCMs with varying edge counts, offering trade-offs between decoding performance and complexity without retraining. Overall, ZeroCode addresses diverse code-design requirements within a unified framework, providing solutions with optimized decoding performance under given constraints.

Abstract: Let $n\ge2$, and let $X=(X_1,\ldots,X_n)$ be a centered Gaussian vector with $\mathrm{Var}(X_i)=1$ for every $i$. Let $Z_1,\ldots,Z_n$ be independent standard Gaussians, and put $\overline{Z}=(Z_1+\cdots+Z_n)/n$. We prove $\mathbb{P}\{\max_i X_i\le t\}\ge\mathbb{P}\{\sqrt{n/(n-1)}\,\max_i(Z_i-\overline{Z})\le t\}$ for every $t\in\mathbb{R}$, and for each fixed $t>0$ equality holds only when $\mathrm{Cov}(X_i,X_j)=-1/(n-1)$ for all $i\ne j$. The right side is the distribution function of the maximum of the regular simplex vector. Equivalently, among all simplices containing a given centered ball, the regular simplex circumscribed about the ball has the least standard Gaussian measure, as conjectured by Balitskiy, Karasev, and Tsigler. In our preceding paper we proved this comparison after both maxima are smoothed by independent Gaussian noise of variance $1/(n-1)$, which suffices for the Weak Simplex Conjecture; here we remove the smoothing, which is what probabilities at a single threshold require. As an application we consider $n$ equally likely signals of equal energy in Gaussian noise, where the transmitter may also send nothing. At every positive false-alarm level, and for every law of a common nonnegative random amplitude not concentrated at zero, the regular simplex uniquely maximizes the average probability of correct identification whenever the signal dimension is at least $n-1$. A Lean formalization is available at https://github.com/abhmul/full-simplex-conjecture-lean.

Abstract: We compare the performance of the RPA and CHIRRUP decoding algorithms for second-order Reed--Muller codes. The RPA algorithm recovers a binary quadratic form associated with a skew-symmetric matrix $M$, whereas CHIRRUP recovers a $\mathbb{Z}_4$-valued quadratic form associated with a symmetric matrix $P$. We describe how the Gray map connects the evaluation vectors of these forms, inducing a rank-preserving correspondence between $M$ and $P$. We analyze how Euclidean distance between evaluation vectors is governed by matrix rank differences using Delsarte--Goethals sets $DG(m,r)$. We demonstrate that CHIRRUP uses repeated measurements more effectively than RPA on these ensembles. While a kernel-aware variant of RPA improves performance on singular quadratic forms by aggregating projection directions across cosets of $\ker M$, CHIRRUP's tree search combines multiple row estimates more efficiently on $DG(m,r)$ when $r < \frac{m-1}{2}$, outperforming vanilla RPA.

Abstract: This paper shows that there are intimate connections between the quadratic transform technique for solving fractional programming (FP) problems and the Schur-complement technique in matrix analysis. We demonstrate that the quadratic transform technique is related to two aspects of the Schur complement: (i) the linear matrix inequality (LMI) condition for positive semidefiniteness and (ii) the matrix determinant formula. Specifically, we establish that the quadratic transform and the Schur-complement LMI condition imply each other. This connection allows us to provide new interpretations of the auxiliary variable in the quadratic transform, and it allows us to rederive the Schur-complement determinant formula. Furthermore, this connection leads to generalizations of the quadratic transform in FP and the Schur-complement LMI that can accommodate generalized matrix inverse. As an application in information theory, we apply the generalized FP framework to the least-favorable-noise minimax formulation of the Gaussian vector broadcast channel sum capacity problem. When the least-favorable noise covariance is singular, matrix-inverse-based Karush-Kuhn-Tucker (KKT) analysis would require a careful analysis of the input and output spaces of the channel. We show using generalized FP that an auxiliary-variable representation of the singular matrix fraction directly yields the reciprocal multiple-access channel and recovers the uplink-downlink duality relation for sum capacity.

Abstract: Given a cyclic code $C_g = \langle g(x) \rangle$ of odd length $m$ over $\mathbb{Z}_4$, one common way to build a quasi-cyclic (QC) code is to pick $f_1, \dots, f_\ell \in \mathbb{Z}_4[x]$ and let $C = \langle (f_1 g, \dots, f_\ell g) \rangle$. Because $\mathbb{Z}_4$ is not a field, the type of $C_g$ ($4^{k_1}2^{k_2}$) is not necessarily inherited by $C$. Determining conditions under which the type of $C_g$ is inherited by $C$ was posed as an open problem recently in \cite{AydinLuOnta2023}. In this paper, we settle this problem. We first give two sufficient conditions for the type to be preserved: one requires a single $f_i$ to be coprime to $x^m-1$ over $\mathbb{F}_2$, the other only requires the $f_i$ to be jointly coprime to it. Neither condition is necessary in general. Using the fact that $x^m-1$ is squarefree for odd $m$, we decompose $\mathbb{Z}_4[x]/\langle x^m-1\rangle$ into a product of finite chain rings using Chinese remainder theorem (CRT) and derive a condition on $f_1, \dots, f_\ell$ that is both necessary and sufficient for $C$ to match the type of $C_g$. This condition depends only on the irreducible factors of $x^m-1$ where $g$ does not already vanish. This also yields a simple test for when $C$ is a free $\mathbb{Z}_4$-module. Finally, we report many new QC codes over $\mathbb{Z}_4$, found by computer searches using Magma software~\cite{Magma1997} guided by this criterion, with Lee distances greater than previously known codes of the same type.

Abstract: We consider an additive white Gaussian noise channel subject to a peak-amplitude constraint and study how the support of the capacity-achieving input distribution (CAID) evolves as the amplitude constraint $A$ varies. Although the CAID is known to be unique, symmetric, discrete, and finitely supported, the structure of its support transitions has remained largely unresolved. We show that the origin is the only possible degenerate support point and the only possible inactive contact point of the KKT function. We then establish continuity properties of the optimal distribution and the KKT function and prove that every nonzero support point is locally stable: under small changes in $A$, it persists as a unique nearby atom whose location and probability mass vary continuously. Combining these results, we prove that, locally, the support cardinality at a nearby amplitude is either unchanged or larger by one and that any such increase can occur only at the origin. Consequently, all local changes in support cardinality are confined to the origin: locally, the only possible transition mechanisms are the appearance or disappearance of the point at the origin and the splitting or merging of the origin into a symmetric pair.