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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 introduce a diffusion framework for neural decoding of error-correcting codes (ECCs). We interpret decoding as a conditional diffusion process that progressively predicts codeword bits while treating the physical channel observation as fixed side information. At each stage, a neural model estimates bitwise posteriors conditioned on the physical observation and the current noisy state of the codeword. We focus particularly on masked diffusion, for which the reverse process naturally induces a Maxwell beam search (MBS) procedure inspired by the Maxwell construction of Measson et al. Rather than committing to a single uncertain decision, the decoder maintains multiple candidate trajectories, removes parity-infeasible candidates, and ranks the remaining candidates using accumulated reverse-transition log-probabilities. We evaluate the framework across multiple code families on binary phase-shift keying (BPSK) over additive white Gaussian noise (AWGN). Experiments show competitive decoding performance across Polar, low-density parity-check (LDPC), and Bose-Chaudhuri-Hocquenghem (BCH) codes. The MBS decoder consistently improves upon the corresponding single-trajectory diffusion decoder and, in all evaluated settings, improves upon available neural decoding 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: This paper investigates the construction of full-systematic permutation codes capable of correcting multiple deletions under two complementary models, namely symbol-invariant deletions (SIDs), where surviving symbol values are preserved, and permutation-invariant deletions (PIDs), where the surviving sequence is standardized to a permutation. For any fixed integer $t \ge 1$ and all sufficiently large message lengths $n$, our proposed encoders map any message permutation of length $n$ to a codeword by inserting distinct redundancy symbols while strictly preserving the sequence order of the original message symbols. The proposed constructions correct up to $t$ deletions using $7t-1$ redundancy markers for PIDs and $4t$ redundancy markers for SIDs, achieving redundancies of $(7t-1)\log n + O_t(1)$ bits and $4t\log n + O_t(1)$ bits, respectively. Both code families are uniformly constructible, encodable, and decodable in $n^{O(t)}$ time. The underlying framework stores an inner deletion-correcting syndrome in the relative positions of redundancy markers via an algebraic outer code based on integer moments and residual graph coloring. We further extend this framework to fixed-composition and strictly $λ$-regular multipermutations, proving that the PID and SID channels coincide whenever the common multiplicity satisfies $λ> t$.