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 $ρ$ of errors in the sum-rank metric with rate $1-ρ-\varepsilon$, for any desired $ρ\in (0,1)$ 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.

💡 Research Summary

The paper addresses a fundamental gap in coding theory for the sum‑rank metric, a distance measure that simultaneously generalizes the Hamming and rank metrics and has become central to modern applications such as multi‑shot network coding, locally repairable codes, space‑time coding, and post‑quantum cryptography. While linearized Reed–Solomon (LRS) codes are known to be maximum‑sum‑rank‑distance (MSRD) and to enjoy many algebraic properties, no explicit families of LRS‑based codes have been shown to support efficient list decoding beyond the unique‑decoding radius. Moreover, folded versions of Reed–Solomon codes have been instrumental in achieving list‑decoding capacity under the Hamming metric, but an analogous theory for folded linearized Reed–Solomon (FLRS) codes in the sum‑rank setting has been missing.

The authors present a unified construction of explicit, (\mathbb{F}_h)-linear subcodes of both LRS and FLRS codes that are provably list‑decodable well beyond the unique‑decoding radius. The key ideas are:

1. Subspace‑design based message restriction.
   The message space (the set of admissible skew‑polynomials) is intersected with a carefully crafted product of subspaces (H_1\times\cdots\times H_k). These subspaces form a subspace design, a combinatorial object guaranteeing that any low‑dimensional periodic affine subspace (the algebraic solution space produced by the decoder) intersects the design in only a few points. The authors use the explicit generalized subspace‑design constructions from Guruswami–Wang–Xing (originally for Gabidulin codes) and adapt them to the sum‑rank setting where the ambient field (\mathbb{F}_h) may be exponentially larger than the block length.

2. Linear‑algebraic list‑decoding framework.
   For a received word (y), an interpolation step builds a skew‑polynomial (Q) that vanishes on all pairs ((\beta_{i,j}, y_{i,j})) corresponding to the received symbols. The vanishing conditions translate into a homogeneous linear system over (\mathbb{F}_h). Solving this system yields an affine subspace (\mathcal{A}) of candidate message vectors. The dimension of (\mathcal{A}) is bounded by (O(s^2/\varepsilon^2)) for LRS and by (O(s)) for FLRS, where (s) is a decoding parameter and (\varepsilon) controls the gap to capacity.

3. Intersection with the subspace design.
   Because both (\mathcal{A}) and the code’s message space are (\mathbb{F}_h)-linear, their intersection can be computed by standard linear‑algebraic operations in time polynomial in the block length and (\log h). The subspace‑design guarantees that this intersection has dimension at most (O(s^2/\varepsilon^2)) (LRS) or (O(s/\varepsilon)) (FLRS), leading to a list size bounded by (h^{O(s^2/\varepsilon^2)}) or (h^{O(s/\varepsilon)}) respectively. A Monte‑Carlo variant using random subspace‑evasive sets further reduces the list size to (O(s/\varepsilon)) with high probability.

The main formal results are:

• Theorem 1.1 (LRS subcodes). For any (\varepsilon\in(0,1)) and integer (s>0), there exists an explicit (\mathbb{F}_h)-linear subcode of an LRS code of rate at least ((1-2\varepsilon)k/n) that can be list‑decoded in polynomial time from up to (s(s+1)(n-k)) sum‑rank errors. The output list lies in an (\mathbb{F}_h)-subspace of dimension (O(s^2/\varepsilon^2)).

• Theorem 1.2 (Folded LRS subcodes). For any (\varepsilon\in(0,1)) and (1\le s\le\lambda), there is an explicit subcode of a (\lambda)-folded LRS code of rate at least ((1-\varepsilon)k/N) (where (N=\lambda n)) that can be list‑decoded from a fraction (\frac{s(s+1)}{n(\lambda-s+1)}) of sum‑rank errors. The list size is at most ((d/\varepsilon)^d) with (d=\lceil k/m\rceil (s-1)=O(s)).

• Theorem 1.3 (Monte‑Carlo construction). Using a random subspace‑evasive set, one can obtain a folded LRS subcode with the same rate and error‑fraction guarantees as in Theorem 1.2, but with list size (O(s/\varepsilon)) with high probability.

All algorithms run in time polynomial in the code length and (\log h); the dominant steps are the interpolation (solving a linear system of size roughly (sn)) and the intersection computation (Gaussian elimination). The paper also provides a detailed analysis of the algebraic structure of the interpolation constraints for folded codes, showing that folding introduces additional linear relations that further shrink the solution space.

From a broader perspective, this work is the first to deliver explicit, positive‑rate sum‑rank metric codes that are efficiently list‑decodable beyond the unique‑decoding radius. It bridges techniques from Hamming‑metric list decoding (folding, subspace designs) with the algebra of skew polynomials that underlie rank‑metric codes. The constructions are fully explicit, avoiding reliance on random coding arguments, and the decoding algorithms are practical for moderate field sizes.

Potential impact areas include:

• Network coding: The ability to correct a larger fraction of sum‑rank errors translates into higher throughput and robustness for multi‑shot network transmissions.
• Locally repairable codes: Subcodes with controlled list size can be used as building blocks for codes with locality constraints while retaining strong error‑correction capabilities.
• Post‑quantum cryptography: Sum‑rank metric codes are candidates for cryptographic primitives; explicit list‑decodable families help assess security margins against decoding attacks.
• Space‑time coding: Folded constructions can be mapped to space‑time block designs, offering improved diversity–multiplexing trade‑offs.

In summary, the paper introduces a novel, algebraically grounded framework that combines subspace‑design restrictions with linear‑algebraic list decoding to achieve the first explicit, efficiently list‑decodable subcodes of LRS and FLRS codes. This advances the theory of sum‑rank metric codes toward practical, capacity‑approaching error correction in a variety of modern communication and cryptographic systems.