Year: 2025
Type: article
Abstract: Linearized Reed–Solomon (LRS) codes are sum-rank-metric codes that generalize both Reed–Solomon and Gabidulin codes. We study vertically and horizontallyinterleavedLRS (VILRS and HILRS) codes whose codewords consist of a fixed number of stacked or concatenated codewords of a chosen LRS code, respectively. Our unified presentation of results for horizontalandvertical 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 either 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 inO(sn2) andÕ(sn2) in most scenarios, respectively, wheresis the interleaving order andndenotes the length of the component code. Errors of sum-rank weight τ =tF+tR+tCconsist oftFfull errors,tRrow erasures, andtCcolumn erasures. Their successful decoding can be guaranteed fortF≤ 1/2 (n−k−tR−tC), 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 whentF≤s/s+1 (n−k−tR−tC) holds for interleaving orders. We give an upper bound on the failure probability for probabilistic unique decoding and showcase its tightness via Monte Carlo simulations.
Authors: Felicitas Hörmann, Hannes Bartz
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416213822
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