Syndrome-Based Error-Erasure Decoding of Interleaved Linearized Reed-Solomon Codes
Felicitas Hörmann, Hannes Bartz
TL;DR
This work addresses robust decoding for interleaved linearized Reed--Solomon codes in the sum-rank metric, presenting unified syndrome-based decoders for both vertical (VILRS) and horizontal (HILRS) interleaving. It develops error-only and error-erasure decoders based on Berlekamp--Massey-like key equations, using ELP for VILRS and ESP for HILRS, and achieves average complexities of $\widetilde{O}(s\,n^2)$ under practical interleaving regimes. The paper derives decoding radii: guaranteed unique decoding up to $\tau \le \tfrac{1}{2}(n-k)$, probabilistic unique decoding up to $\tau \le \tfrac{s}{s+1}(n-k)$, and analogous bounds in the error-erasure setting, with explicit probability bounds and extensive simulations. It also introduces efficient subroutines—multisequence skew-feedback shift-register synthesis, Skachek--Roth-like root finding, and Gabidulin-like linear-algebra solves—to realize scalable decoding. The results enable practical, high-rate, multishot network coding and cryptographic applications by enabling reliable decoding of interleaved sum-rank codes with controlled failure probability and polynomial-time complexity.
Abstract
Linearized Reed--Solomon (LRS) codes are sum-rank-metric codes that generalize both Reed--Solomon and Gabidulin codes. We study vertically and horizontally interleaved LRS (VILRS and HILRS) codes whose codewords consist of a fixed number of stacked or concatenated codewords of a chosen LRS code. Our unified presentation of results for horizontal and vertical interleaving is novel and simplifies the recognition of resembling patterns. This paper's main results are syndrome-based decoders for both VILRS and HILRS codes. We first consider an error-only setting and then present more general error-erasure decoders, which can handle full errors, row erasures, and column erasures simultaneously. Here, an erasure means that parts of the row space or the column space of the error are already known before decoding. We incorporate this knowledge directly into Berlekamp--Massey-like key equations and thus decode all error types jointly. The presented error-only and error-erasure decoders have an average complexity in $O(sn^2)$ and $\widetilde{O}(sn^2)$ in most scenarios, where $s$ is the interleaving order and $n$ denotes the length of the component code. Errors of sum-rank weight $τ=t_{\mathcal{F}}+t_{\mathcal{R}}+t_{\mathcal{C}}$ consist of $t_{\mathcal{F}}$ full errors, $t_{\mathcal{R}}$ row erasures, and $t_{\mathcal{C}}$ column erasures. Their successful decoding can be guaranteed for $t_{\mathcal{F}}\leq\tfrac{1}{2}(n-k-t_{\mathcal{R}}-t_{\mathcal{C}})$, where $n$ and $k$ represent the length and the dimension of the component LRS code. Moreover, probabilistic decoding beyond the unique-decoding radius is possible with high probability when $t_{\mathcal{F}}\leq\tfrac{s}{s+1}(n-k-t_{\mathcal{R}}-t_{\mathcal{C}})$ holds for interleaving order $s$. We give an upper bound on the failure probability for probabilistic unique decoding and showcase its tightness via Monte Carlo simulations.