The sum-rank metric is a hybrid between the Hamming metric and the rank metric and suitable for error correction in multishot network coding and distributed storage as well as for the design of quantum-resistant cryptosystems. In this work, we consider the construction and decoding of folded linearized Reed-Solomon (FLRS) codes, which are shown to be maximum sum-rank distance (MSRD) for appropriate parameter choices. We derive an efficient interpolation-based decoding algorithm for FLRS codes that can be used as a list decoder or as a probabilistic unique decoder. The proposed decoding scheme can correct sum-rank errors beyond the unique decoding radius with a computational complexity that is quadratic in the length of the unfolded code. We show how the error-correction capability can be optimized for high-rate codes by an alternative choice of interpolation points. We derive a heuristic upper bound on the decoding failure probability of the probabilistic unique decoder and verify its tightness by Monte Carlo simulations. Further, we study the construction and decoding of folded skew Reed-Solomon codes in the skew metric. Up to our knowledge, FLRS codes are the first MSRD codes with different block sizes that come along with an efficient decoding algorithm.

翻译：和秩度量是汉明度量与秩度量的混合形式，适用于多跳网络编码、分布式存储中的纠错，以及抗量子密码系统的设计。本文研究了折叠线性化Reed-Solomon（FLRS）码的构造与译码问题，通过适当参数选择可证明其为最大和秩距离（MSRD）码。我们推导了一种基于插值的FLRS码高效译码算法，该算法既可作为列表译码器，也可作为概率唯一译码器。所提出的译码方案能够纠正超过唯一译码半径的和秩错误，其计算复杂度与未折叠码长呈二次关系。我们展示了如何通过选择替代插值点来优化高码率码的纠错能力，推导了概率唯一译码器译码失败概率的启发式上界，并通过蒙特卡洛仿真验证了其紧致性。此外，我们还在斜度量下研究了斜Reed-Solomon码折叠变体的构造与译码。据我们所知，FLRS码是首个具有不同分块大小且配备高效译码算法的MSRD码。