Kuo Shang | Searcharxiv

arXiv2026

Explicit List-Decodable Linearized Reed-Solomon and Folded Linearized Reed-Solomon Subcodes

The sum-rank metric is the mixture of the Hamming and rank metrics. The sum-rank metric found its application in network coding, locally repairable codes, space-time coding, and quantum-resistant cryptography. Linearized Reed-Solomon (LRS) codes are the sum-rank analogue of Reed-Solomon codes and strictly generalize both Reed-Solomon and Gabidulin codes. In this work, we construct an explicit family of $\mathbb{F}_h$-linear sum-rank metric codes over arbitrary fields $\mathbb{F}_h$. Our construction enables efficient list decoding up to a fraction $\rho$ of errors in the sum-rank metric with rate $1-\rho-\varepsilon$, for any desired $\rho \in (0,1)$ and $\varepsilon>0$. Our codes are subcodes of LRS codes, obtained by restricting message polynomials to an $\mathbb{F}_h$-subspace derived from subspace designs, and the decoding list size is bounded by $h^{\mathrm{poly}(1/\varepsilon)}$. Beyond the standard LRS setting, we further extend our linear-algebraic decoding framework to folded Linearized Reed-Solomon (FLRS) codes. We show that folded evaluations satisfy appropriate interpolation conditions and that the corresponding solution space forms a low-dimensional, structured affine subspace. This structure enables effective control of the list size and yields the first explicit positive-rate FLRS subcodes that are efficiently list decodable beyond the unique-decoding radius. To the best of our knowledge, this also constitutes the first explicit construction of positive-rate sum-rank metric codes that admit efficient list decoding beyond the unique decoding radius, thereby providing a new general framework for constructing efficiently decodable codes under the sum-rank metric.

cs.IT↗

arXiv2024

Randomness-Efficient Constructions of Capacity-Achieving List-Decodable Codes

We wish to generate list-decodable codes over small alphabets using as little randomness as possible. Specifically, we hope to generate codes achieving what we term the Elias bound, which means that they are $(\rho,L)$-list-decodable with rate $R \geq 1-h(\rho)-O(1/L)$. A long line of work shows that uniformly random linear codes (RLCs) achieve the Elias bound: hence, we know $O(n^2)$ random bits suffice. Prior works demonstrate that just $O(Ln)$ random bits suffice, via puncturing of low-bias codes. These recent constructions are combinatorial. We provide two new constructions, which are algebraic. Compared to prior works, our constructions are simpler and more direct. Furthermore, our codes are designed in such a way that their duals are also quite easy to analyze. Our first construction -- which can be seen as a generalization of the Wozencraft ensemble -- achieves the Elias bound and consumes $Ln$ random bits. Additionally, its dual code achieves the GV-bound with high probability, and both the primal and dual admit quasilinear-time encoding algorithms. The second construction consumes $2nL$ random bits and yields a code where both it and its dual achieve the Elias bound. As we discuss, properties of a dual code are often crucial for applications in cryptography. In all of the above cases -- including the prior works achieving randomness complexity $O(Ln)$ -- the codes are designed to "approximate" RLCs. Namely, for a given locality parameter $L$ we construct codes achieving the same $L$-local properties as RLCs. This allows one to appeal to known list-decodability results for RLCs and thereby conclude that the code approximating an RLC also achieves the Elias bound. As a final contribution, we indicate that such a proof strategy is inherently unable to generate list-decodable codes of rate $R$ over $\mathbb F_q$ with less than $L(1-R)n\log_2(q)$ bits of randomness.

cs.IT↗