Efficient Decoding of Folded Linearized Reed-Solomon Codes in the Sum-Rank Metric

Abstract

Recently, codes in the sum-rank metric attracted attention due to several applications in e.g. multishot network coding, distributed storage and quantum-resistant cryptography. The sum-rank analogs of Reed-Solomon and Gabidulin codes are linearized Reed-Solomon codes. We show how to construct $h$-folded linearized Reed-Solomon (FLRS) codes and derive an interpolation-based decoding scheme that is capable of correcting sum-rank errors beyond the unique decoding radius. The presented decoder can be used for either list or probabilistic unique decoding and requires at most $\mathcal{O}(sn^2)$ operations in $\mathbb{F}_{q^m}$, where $s \leq h$ is an interpolation parameter and $n$ denotes the length of the unfolded code. We derive a heuristic upper bound on the failure probability of the probabilistic unique decoder and verify the results via Monte Carlo simulations.