Error-Erasure Decoding of Linearized Reed-Solomon Codes in the Sum-Rank Metric

TL;DR

This paper introduces the first error-erasure decoding algorithm for Linearized Reed-Solomon codes in the sum-rank metric, enabling improved error correction in multishot network coding and related applications.

Contribution

It presents a syndrome-based Berlekamp-Massey-like decoder for LRS codes that corrects errors and erasures efficiently, expanding their practical utility.

Findings

Decodes errors and erasures with complexity O(n^2) in finite fields.

Corrects up to 2t_F + t_R + t_C ≤ n - k in sum-rank metric.

Applicable to sum-subspace metric in multishot network coding.

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 $t_F$ full errors, $t_R$ row erasures and $t_C$ column erasures up to $2t_F + t_R + t_C \leq n-k$ in the sum-rank metric requiring at most $\mathcal{O}(n^2)$ operations in $\mathbb{F}_{q^m}$, 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.