Error-Erasure Decoding of Linearized Reed-Solomon Codes in the Sum-Rank Metric
(2202.06758)Abstract
Codes in the sum-rank metric have various applications in error control for multishot network coding, distributed storage and code-based cryptography. Linearized Reed-Solomon (LRS) codes contain Reed-Solomon and Gabidulin codes as subclasses and fulfill the Singleton-like bound in the sum-rank metric with equality. We propose the first known error-erasure decoder for LRS codes to unleash their full potential for multishot network coding. The presented syndrome-based Berlekamp-Massey-like error-erasure decoder can correct $tF$ full errors, $tR$ row erasures and $tC$ column erasures up to $2tF + tR + tC \leq n-k$ in the sum-rank metric requiring at most $\mathcal{O}(n2)$ operations in $\mathbb{F}_{qm}$, where $n$ is the code's length and $k$ its dimension. We show how the proposed decoder can be used to correct errors in the sum-subspace metric that occur in (noncoherent) multishot network coding.
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