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.
翻译：超新度中的代码在多发网络编码、分布式存储和代码加密的错误控制中有各种应用。 线性 Reed- Solomon (LRS) 代码包含作为子类的 Reed- Solomon 和 Gabidulin 代码, 并实现Singon- 类似Single 约束的单吨代码。 我们为 LRS 代码提出了第一个已知的错误极限解码器, 以释放其在多发网络编码中的全部潜力。 以 $\ mathbb{ F ⁇ q ⁇ m} 提供的基于 Insmex- game- massey 类似错误- 系统解码器可以纠正$t_ F 全误差, $t_ R$ 的行缩放和$t_ C专列删除到$t_ F + t_ R + t_ c t_ c\leq n- k$ 。 我们展示了如何在最需要 $\ mathcal=mathb{F\\\\\q\\\\ $ $, 其中, $n$n$n$n$n$n$rage com- small in most decodecodespacedespace commall commal
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