communication engineers

Abstract: Ensemble decoding improves decoding reliability by combining component decoders with complementary failure events. Existing approaches typically obtain this diversity from code-specific structure, such as code automorphisms, multiple parity-check representations, or targeted modifications of the message-passing graph. In this work, we instead learn the diversity of an ensemble of weighted belief propagation decoders by jointly optimizing trainable Tanner-graph edge weights. A central design question is how the training objective should distribute gradient among the ensemble members to promote complementary decoding behavior. To this end, we compare four ensemble objectives under direct gradient-based optimization and identify best-member training as the most effective approach. For a (64,32) progressive edge-growth (PEG) low-density parity-check (LDPC) code, the resulting ensemble outperforms the considered trained and untrained belief-propagation-based ensemble baselines. With 10 members, it provides a gain of approximately 0.66 dB over a single belief propagation decoder at a target frame error rate (FER) of $10^{-4}$. Similar behavior is observed for a (128,64) CCSDS LDPC code, indicating that the learned complementarity is not specific to PEG. Although the ensemble performance improves substantially, all trained members show a higher standalone FER. The gain therefore results from learned specialization and complementary failure events rather than uniformly improved component decoders.

Abstract: We extend the complete classifications of linear complementary dual (LCD) codes from length 15 to length 19 in the binary case, from length 10 to length 15 in the ternary case and from length 12 to length 13 in the quaternary Hermitian case. The classification rests on a reduction of the monomial equivalence of LCD codes to graph isomorphism on the coordinates, combined with canonical augmentation.

Abstract: The $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive codes are subgroups of $\mathbb{Z}_p^{α_1}\times\mathbb{Z}_{p^2}^{α_2}$, and they can be seen as linear codes over $\mathbb{Z}_p$ when $α_2=0$, as $\mathbb{Z}_{p^2}$-additive codes when $α_1=0$, and as $\mathbb{Z}_2\mathbb{Z}_4$-additive codes when $p=2$. A $\mathbb{Z}_p\mathbb{Z}_{p^2}$-linear generalized Hadamard (GH) code is a GH code over $\mathbb{Z}_p$ which is the Gray map image of a $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive code. For every prime $p$, there are recursive constructions of $\mathbb{Z}_p\mathbb{Z}_{p^2}$-linear GH codes $H_p^{t_1,t_2}$ with $α_1\neq0$ and $α_2\neq0$, and of $\mathbb{Z}_{p^s}$-linear GH codes $\bar H_p^{t_1,\dots,t_s}$. We show that the kernel blocks of a nonlinear code $H_p^{t_1,t_2}$ give equivalent punctured codes, and that these punctured codes are linear. Any of them is called the derived code of $H_p^{t_1,t_2}$. We also determine the ranks of the codes $H_p^{t_1,t_2}$ and $\bar H_p^{t_1,t_2}$. For odd $p$, the rank of $H_p^{t_1,t_2}$ depends on the Bernoulli numbers modulo $p$. By comparing the derived codes and the ranks, we show that, for odd $p$, no code $H_p^{t_1,t_2}$ is equivalent to any of these $\mathbb{Z}_{p^s}$-linear GH codes. For $p=2$, a nonlinear code $H_2^{t_1,t_2}$ is equivalent to a $\mathbb{Z}_{2^s}$-linear Hadamard code if and only if $t_2\geq2$. Finally, we count the nonequivalent codes in the union of both families.

Abstract: We prove that every nontrivial linear $[n,k]$ maximum distance separable (MDS) code over a finite field $\F_q$ of characteristic two satisfies $n\le q+1$, except in dimensions $k=3$ and $k=q-1$, where $n\le q+2$. This resolves the linear MDS conjecture in even characteristic.

Abstract: We study structural properties of polynomial ideal (PI) codes and additive Reed--Solomon (aRS) codes in the folded Hamming metric (equivalently, the Hamming metric over a larger alphabet). We first determine the maximum possible length of PI codes, prove that the dual of every PI code is a generalized PI code, and deduce that PI codes are dually QMDS together with an explicit formula for their weight distribution. We then show that PI codes defined by irreducible polynomials correspond to aRS codes with pairwise nonconjugate evaluation points. For these aRS codes, the previous results on PI codes apply (duality, QMDS property and weight distribution). We show that, asymptotically, aRS codes with nonconjugate evaluation points achieve the same length as PI codes. We also compute the permutation group of such aRS codes. We also extend the family of PI codes to consider arbitrary polynomial modules (not necessarily pairwise coprime), and study when these and general aRS codes satisfy the QMDS and dually QMDS properties.

Abstract: 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.

Abstract: Linear complementary dual (LCD) codes are linear codes that intersect their dual trivially. We study double circulant (Euclidean) LCD codes over the finite chain ring $R=\mathbb F_2+u\mathbb F_2$, where $u^2=0$. Using the Chinese Remainder Theorem and the factorization of $x^n-1$, we derive an explicit enumeration formula, with a separate treatment of the constituent corresponding to $x-1$, which is not obtained by a formal specialization of the previously known counting formula over $\mathbb F_q[u]/(u^k)$. Assuming Artin's primitive root conjecture for $2$, we show that the binary Gray images contain an asymptotically good family of rate $1/2$ whose relative distance can approach $H_2^{-1}(1/4)$ from below. More importantly from the constructive viewpoint, by combining Gray images with a known extension construction, we obtain a binary LCD code with parameters $[29,15,7]$ and construct two explicit permutation-inequivalent binary LCD codes with parameters $[45,23,9]$.

Abstract: Levenshtein's classical theorem shows that any code that can uniquely decode from \(t\) deletions can also uniquely decode from \(t\) insertions. This symmetry, however, does not generally hold in the list-decoding regime. Several prior results on the capacity and radii of list-decoding from insertions and deletions suggest that insertions are easier to handle than deletions. In an effort to better understand this asymmetry between insertions and deletions under list-decoding, we study an analogous version of McEliece's \emph{average list size problem}. More precisely, given a binary code \(\mathcal{C}\) of length \(n\) that uniquely corrects \(t'\) insertions (or deletions), we are interested in the average number of codewords returned by a \(t\)-insertion (or deletion) list decoder, where \(t' < t\). Focusing on the case of \(t'=1\) and \(t=2\), we find that under two insertions, this quantity is at most \((n+12)/4\) for any VT code of length \(n\), whereas under two deletions it is at least \(n/2 - 10\) when averaged over all length-\(n\) VT codes. The nearly factor-of-two gap between these upper and lower bounds serves to confirm the relative ease of list-decoding of insertions compared to deletions from an average-case perspective.

Abstract: We study the communication cost of exact sampling under Campbell's exponential coding criterion. Given a target distribution $P$ and a proposal distribution $Q$, the sender and receiver share i.i.d. $Q$-distributed samples, and the sender communicates an index so that the selected sample has distribution exactly $P$. We introduce a new information measure Rényi channel simulation divergence $R_α(P\Vert Q)$, a Rényi generalization of the channel simulation divergence. We then show that it lower bounds the Campbell cost of every exact sampler and is achievable, via a new layered sampling scheme, within an additive constant depending only on the Campbell parameter. This yields a tighter one-shot characterization than previous bounds based on Rényi divergences of different orders. For i.i.d. product distributions, under suitable non-degeneracy and moment conditions, we further show that the gap beyond the leading Rényi-divergence term is $\frac{1}{2}\log n+O(1)$.

Abstract: Error-correcting codes are a foundational technology for representing information robustly against noise in communication and data storage. The binary code packing problem, which seeks to maximize the number of codewords at a fixed code length and minimum distance, is a fundamental problem in coding theory that remains open for many parameter choices. We give explicit binary codes with parameters $(n,M,d)=(22,84,9)$, $(24,196,9)$, and $(25,65,11)$, where $n$, $M$, and $d$ denote the code length, number of codewords, and minimum distance, respectively. These establish the lower bounds $A_2(22,9)\ge84$, $A_2(24,9)\ge196$, and $A_2(25,11)\ge65$, where $A_2(n,d)$ denotes the maximum possible number of codewords, improving the respective lower bounds 80, 192, and 64 in the public table of general binary codes. Most notably, the third result improves the lower bound supplied by a lexicographic construction for the first time in 66 years, while the first two improve bounds that have stood for 21 years. The new codes represent more messages without changing the code length or error-correction guarantees, with gains compounding multiplicatively across blocks. We release all codewords and an exhaustive verifier to enable independent verification.

Abstract: We develop q-ary Guessing Random Additive Noise Decoding (GRAND) for linear codes over a q-ary alphabet. The decoder works on a sorted symbol-likelihood array and uses three local child generation rules to generate symbol deviation patterns. We then prove that these rules induce a monotone rooted spanning tree of the full row-index space, so best-first traversal gives maximum-likelihood (ML) decoding under unlimited search. Reed-Solomon simulations verify ML agreement and show the finite-budget performance-complexity tradeoff.

Abstract: Understanding the limits of list-decodability of Reed-Solomon codes has been one of the most important open problems in algebraic coding theory. Recently, Brakensiek, Chen, Putterman, Zhang, and Zheng, in a remarkable breakthrough, showed that Reed-Solomon (RS) codes over fields of large characteristic are algorithmically list-decodable all the way up to capacity. Building on this result, Jeronimo subsequently extended these techniques to solve the proximity-gaps question for RS codes. In this article, we give a unified and transparent exposition of these results.

Abstract: A $k$-substring edit in a sequence first deletes a substring of length at most $k$, and then inserts a sequence of length at most $k$ at the same position. A code that can correct a $k$-substring edit is called a $k$-substring edit code. In this paper, we develop a new localization method and use it to construct a $q$-ary $k$-substring edit correcting code with $\log n+8\log\log n+o(\log\log n)$ bits of redundancy for any fixed $q\ge2$ and $k\ge1$, where $n$ is the code length. For the binary alphabet, this improves upon the redundancy $\log n+16k\log\log n+o(\log\log n)$ obtained by Li \emph{et al}. When the deleted substring and the inserted sequence have different lengths, we further construct codes with redundancy $\log n+O_{q,k}(1)$, which is optimal up to an additive constant. As corollaries, for all fixed $q\ge2$ and $1\le t\le T$, we obtain $q$-ary $(\le t)$-burst-deletion correcting codes and $(t,T)$-localized deletion correcting codes with redundancies $\log n+O_{q,t}(1)$ and $\log n+O_{q,T}(1)$, respectively. To the best of our knowledge, these are the first constructions attaining optimal redundancy up to an additive constant for these two deletion models over the full range of fixed parameters. For $(\le t)$-burst-deletion correction with $t\ge2$, such redundancy had previously been achieved only for $q=t=2$ by Levenshtein in 1967.

Abstract: In error correction, the difference between a codeword and a received word, a candidate noise sequence, carries information on the confidence of the codeword as a proposed decoding. For additive channels, it is known that if noise sequences are ranked in decreasing order of likelihood, a lower rank correlates with higher confidence in a decoding. Here we substantially expand considerations by exploring the relationship between the likelihood ranks arising in list decoding and the posterior probability of decoding correctness, establishing that the ranks associated with subsequent list entries carry reliability information that is not obtainable from the first rank alone. Building on the recent development of Soft-Output Guessing Random Additive Noise Decoding, we first show that, under a random codebook model, soft-output expressions for list decoding coincide with the exact finite-blocklength conditional probabilities of decoding correctness. For hard-decision maximum likelihood decoding, we characterize the asymptotic reliability of the first list entry. We identify a decision function whose sign determines whether the posterior correctness probability of the first entry in the list converges to one or zero and whose absolute value gives the exponential rate of convergence. Perhaps surprisingly, the asymptotic reliability of the first list entry depends only on the first, second and last ranks. For list size one, the result recovers the previously known likelihood-rank threshold for reliable decoding. For list size two, a sufficiently late second entry can imply that the first entry is asymptotically correct even when its rank, viewed in isolation, would indicate otherwise.

Abstract: The kissing-number problem is a classical problem in discrete geometry whose exact solution is known in only a few dimensions. Recent artificial-intelligence approaches have begun to discover improved configurations through large-scale numerical and combinatorial search, but converting such searches into general mathematical constructions and rigorous proofs remains challenging. Here we use Qiushi Engine, an autonomous multi-agent research system, to investigate kissing numbers and obtain new lower bounds in nineteen dimensions: $25$, $27$, $32$--$39$, $43$, $45$, and $49$--$55$. The resulting constructions arise from distinct structural mechanisms, including coordinated motions of contact layers, labelled direction reuse, joint support exchanges, signed-code replacements, cross-shell lattice constructions, low-overlap lattice isometries, and spherical-design moment certificates. They yield sharp capacities for parameterized signed-code models, deterministic image-union guarantees, and exact section and projection counts controlled by embedded root systems and anchor-graph statistics. These methods yield, among others, $K(25)\ge197580$, $K(27)\ge201567$, $K(38)\ge591900$, $K(43)\ge2553792$, $K(45)\ge7380090$, and $K(55)\ge53301140$. The autonomous system carried out the construction searches, mathematical analysis and computational verification, while each final result was reduced to explicit mathematical arguments and independently checkable finite certificates. Our results illustrate how autonomous research systems can move beyond optimization within fixed formulations to discover new mathematical representations and constructions at scale.

Abstract: We study the maximum sum rate of uniquely decodable code pairs. A weighted refinement of a complement-gluing construction yields an explicit code of length 672 and sum rate exceeding $1.318639029203$. An iterated projection argument, combined with a conditional entropy estimate and a justified continuous limit, gives an upper bound of $1.480063425539$.

Abstract: Hyun, Kim, Wu and Yue constructed optimal and minimal binary linear codes from order ideals of hierarchical posets with two levels. Two different generalizations of the underlying antichain (simplicial complex) setting to odd characteristic are known: down-sets of $\mathbb{F}_p^n$ under the componentwise order, and support-closed subsets of $\mathbb{F}_q^m$. No generalization of the poset setting itself has appeared. We introduce generalized order ideals of a poset of order $p-1$, obtained by attaching multiplicities in ${0,\dots,p-1}$ to the elements of a poset, and study the two natural notions of order ideal that arise for hierarchical posets with two levels. Whenever the ideal meets the upper level, the resulting defining sets are neither down-sets nor support-closed. We determine the weight distributions of the associated complement codes, exhibit a family of Griesmer codes in which the upper element carries an arbitrary multiplicity, and, via the characteristic function of a generalized order ideal, obtain an infinite family of minimal $p$-ary codes of length $p^n-1$ and dimension $n+1$ violating the Ashikhmin--Barg condition.

Abstract: Networks of interacting Hebbian networks have recently been shown to perform a task beyond associative memory, namely \emph{pattern disentanglement}: when fed with a spurious mixture of stored patterns, the different modules spontaneously specialize on, and retrieve, the different constituents of the mixture. So far, this capability has only been established for pairwise interactions, which limits the number of patterns that can be handled. Here we introduce a dense extension of these modular networks, in which both the auto-associative couplings within each module and the hetero-associative couplings among modules are promoted to higher-order Hebbian interactions. We show that, with a suitable choice of the interaction orders, the network disentangles mixtures while storing a number of patterns that scales linearly with the module size, a regime where its pairwise counterpart fails. Through a statistical-mechanical analysis based on Guerra's interpolation, we derive the self-consistency equations for the order parameters and draw the phase diagrams identifying the region where disentanglement is achieved; these predictions are confirmed by Monte Carlo simulations. Finally, we show that disentanglement provides a natural decoding primitive, and we illustrate it with two applications: the explicit reconstruction of all the hidden patterns from the Hebbian tensors and a stream of unlabeled mixtures, and a proof-of-concept communication protocol in which each message token is transmitted as a masked mixture of hidden patterns and decoded by the network dynamics. Owing to its attractor-based decoding, the protocol degrades gracefully under strong channel corruption, where conventional secure-transmission pipelines fail abruptly.

Abstract: Let \(X=(X_1,\ldots,X_n)\) be uniform on the Boolean cube, and let \(Y\) be obtained by passing its coordinates independently through a binary symmetric channel with crossover probability \(p\). For a \(k\)-bit quantizer \(Q(X)\), reporting \(k\) input coordinates retains \(k(1-h_2(p))\) bits of information, making coordinate projections a natural quantization benchmark. For \(k=1\), this is the Courtade--Kumar conjecture: every Boolean function \(f\) satisfies \(I(f(X);Y)\leq 1-h_2(p)\), with equality for coordinate functions. We prove the conjecture in every dimension and for every crossover probability, without a balance assumption. The proof tracks the information gap relative to a coordinate as the channel correlation varies. After a monotone rearrangement, an entropy-flow identity relates the derivative of this gap to the edge boundary of the associated Boolean decision set. A sharp local entropy comparison, Fourier analysis, and logarithmic Sobolev estimates for pivotal sets imply that any positive gap must grow as the noise decreases, contradicting its nonpositive value at the noiseless endpoint. For \(k\geq8\), a coding-theoretic phenomenon emerges. We construct a shortened-Hamming-code quantizer that beats the coordinate benchmark for \(k=8\), and a direct-sum argument extends the failure to every \(k\geq8\). Thus structured coding can retain more information than coordinate selection. Exhaustive searches within several structured code families and Monte Carlo searches over random codebooks for \(2\leq k\leq7\) find that structured codebooks outperform random ones but remain below the coordinate benchmark. This motivates our conjecture that coordinate projections are optimal for \(2\leq k\leq7\).

Abstract: We prove the Courtade-Kumar conjecture: among all Boolean functions $f\colon \{-1,1\}^n\to\{-1,1\}$, a dictator retains the most information about a uniformly random input observed through independent binary noise.

Abstract: Determining the exact minimum distances of BCH codes remains a open problem. We establish the minimum distances of several families of primitive narrow-sense BCH codes, showing that they attain their designed distances. Our approach centers on $\mathbb{F}_q$-good zero-sets, which we introduce through a derivative condition on their vanishing polynomials. We show that a $q$-ary primitive narrow-sense BCH code of length $q^m-1$ and designed distance $2\leqδ\leq q^m-1$ has minimum distance $δ$ if and only if there exists an $\mathbb{F}_q$-good zero-set of cardinality $δ+1$ in the finite field $\mathbb{F}_{q^m}$ with $q^m$ elements. To construct $\mathbb{F}_q$-good zero-sets, we develop several methods based on polynomial substitutions, power maps, and shifted inverses, as well as direct constructions using polynomials of special forms. Together with suitable initial $\mathbb{F}_q$-good zero-sets, including those arising from known minimum-distance results, these methods yield new good zero-sets of various cardinalities and hence families of primitive narrow-sense BCH codes whose minimum distances equal their designed distances. These families cover a broad range of designed distances, with several known minimum-distance results recovered as special cases.

Abstract: The sequence reconstruction problem for $q$-ary deletion channels, introduced by Levenshtein in 2001, concerns the minimum number of channels required to uniquely recover a transmitted sequence when each channel introduces exactly $t$ deletions. Combinatorially, it is equivalent to determining $N_q(n,d,t)$, the maximum intersection size of two $t$-deletion balls with centers at Levenshtein distance at least $d$, for $q$-ary sequences of length $n$ over the alphabet \(Σ_q=\{0,1,\dots,q-1\}\). Levenshtein solved the uncoded case $N_q(n,1,t)$ for all $n\ge t$; subsequently, Gabrys and Yaakobi determined $N_2(n,2,t)$, and Wang et al. extended the result to $N_3(n,2,t)$. In this paper, we study the problem for \(q\)-ary sequences under minimum Levenshtein distance \(d=2\) with channels that introduce exactly \(t\) deletions. We determine the exact value of \(N_q(n,2,t)\) for all \(t\ge 2, q\geq 4\), and for sufficiently large \(n\), and construct explicit pairs of sequences attaining the maximum intersection. Furthermore, for each $q\ge3$, we characterize all extremal sequence pairs. In particular, if the intersection size matches the first two terms of \(N_q(n,2,t)\), then the two center sequences must contain, at the same positions, length-5 blocks of the forms \((a,b,c,a,b)\) and \((b,a,c,b,a)\) for some distinct \(a,b,c\inΣ_q\); for \(t\ge q+2\), the exact maximum \(N_q(n,2,t)\) is attained precisely by \(2q!\) unordered pairs of sequences with a specific block structure. Asymptotically, we prove that for \(q\ge 4\) and \(t\ge 2\), \[ N_q(n,2,t)=\frac{6}{(t-2)!}n^{t-2}-\frac{3t+13}{(t-3)!}n^{t-3}+\frac{3t^2+25t+64}{4(t-4)!}n^{t-4}+O(n^{t-5}). \] Moreover, \(N_q(n,2,t)\) and \(N_{q-1}(n,2,t)\) share their first \(q-1\) terms, and for \(t\ge q\) the coefficient of \(n^{t-q}\) in their difference is \(\frac{6t-6q+5}{(t-q)!}\).

Abstract: We study binary quasi-cyclic LDPC codes whose parity-check matrices are full arrays of single circulant permutation matrices (CPMs), referred to here as CPM-LDPC codes. Their minimum distance is at most $(J+1)!$, where $J$ is the column weight. For every fixed pair of column and row weights $2\le J<L$, we show that this bound is attained for all sufficiently large integer lift sizes. First, we give one integer exponent matrix independent of the lift size $P$. Second, we show that independent uniform exponent choices attain the bound with probability $1-O_{J,L}(P^{-1})$. Both proofs use cycle conditions required by low-weight codewords and a lower bound on the number of terms in vectors satisfying polynomial check equations. Neither construction requires $P$ to be prime. We also give small-lift arrays attaining the bound 24 for $J=3$, $L=4,\ldots,8$, and arrays with distance at least 28 for $J=4$, $L=5,\ldots,8$, together with computational distance verification.

Abstract: We consider channels with additive colored Gaussian noise and noiseless feedback. Kim's seminal work derived a stationary variational characterization of feedback capacity and further asserted that the capacity-achieving stationary input need not contain a feedback-independent Gaussian component. These results led to the construction of a simple coding scheme, based on the Schalkwijk--Kailath (SK) refinement principle, which was shown to be capacity-achieving. A recent note identified a gap in the proof of the feedback-independent component-removal assertion, thereby leaving the optimality of the SK scheme and subsequent results that rely on it incomplete. In this paper, we prove the component-removal assertion for channels with stationary Gaussian noise that has a rational power spectral density. Our proof uses a perturbation analysis of a convex optimization formulation of feedback capacity and, indeed, shows that every optimizer assigns zero power to the feedback-independent component. Using this stronger property, we construct from any optimizer an explicit SK coding scheme that achieves every rate below feedback capacity with doubly-exponentially decaying maximal error probability.

Abstract: In recent years, simplicial complexes have gained considerable attention as a useful tool for constructing distance-optimal codes over finite fields. In this article, we construct four infinite families of linear codes over the ring $\mathcal{R}=\mathbb{F}_{q}+u\mathbb{F}_{q}$ with $u^2=0$ using simplicial complexes with one or two maximal elements, and completely determine their Lee weight distributions via exponential-sum techniques. By employing a Gray map on $\mathcal{R}$, we obtain infinite families of distance-optimal codes over $\mathbb{F}_{q}$, including a near-Griesmer family, and establish sufficient conditions for their minimality. Furthermore, we investigate the corresponding subfield codes and derive sufficient conditions for their distance-optimality and minimality, yielding infinite families of Griesmer and near-Griesmer codes.

Abstract: Affine Grassmann codes, introduced by Beelen, Ghorpade and Høholdt, are linear codes over $\mathbb{F}_q$ obtained by evaluating linear combinations of minors of a generic matrix. The weight spectrum of the affine Grassmann code $C^{\mathbb{A}}(2,m)$ was determined by Piñero and Singh via a case analysis on the rank of an associated alternating matrix. We give an independent and more streamlined derivation, valid for all $m\ge4$ and every prime power $q$, in which each codeword is written in a compact matrix form and its Hamming weight is expressed through a single closed formula involving two explicit affine subspaces and their intersection.

Abstract: We give a deterministic polynomial-time list-decoding algorithm for Reed-Solomon codes over prime fields that approaches list-decoding capacity for every evaluation set and every constant rate.