We construct $s$-interleaved linearized Reed--Solomon (ILRS) codes and variants and propose efficient decoding schemes that can correct errors beyond the unique decoding radius in the sum-rank metric. The proposed interpolation-based scheme for ILRS codes can be used as a list decoder or as a probabilistic unique decoder that corrects errors of sum-rank up to $t\leq\frac{s}{s+1}(n-k)$, where $s$ is the interleaving order, $n$ the length and $k$ the dimension of the code. Upper bounds on the list size and the decoding failure probability are given where the latter is based on a novel Loidreau--Overbeck-like decoder for ILRS codes. We show how the proposed decoding schemes can be used to decode errors beyond the unique decoding radius in the skew metric by using an isometry between the sum-rank metric and the skew metric. We generalize fast minimal approximant basis interpolation techniques to obtain efficient decoding schemes for ILRS codes (and variants) with subquadratic complexity in the code length. Up to our knowledge, the presented decoding schemes are the first being able to correct errors beyond the unique decoding region in the sum-rank and skew metric. The performance of the proposed decoding schemes and the tightness of the upper bound on the decoding failure probability are validated via Monte Carlo simulations.

翻译：本文构造了$s$交错线性化Reed-Solomon（ILRS）码及其变种，并提出了高效的译码方案，可在和秩度量下纠正超出唯一译码半径的差错。所提出的基于插值的ILRS码方案可作为列表译码器或概率唯一译码器使用，能纠正和秩不超过$t\leq\frac{s}{s+1}(n-k)$的差错，其中$s$为交错阶数，$n$为码长，$k$为码的维数。给出了列表大小与译码失败概率的上界，其中后者基于一种新颖的ILRS码类Loidreau-Overbeck译码器。我们展示了如何通过利用和秩度量与斜度量之间的等距映射，将所提译码方案用于纠正斜度量下超出唯一译码半径的差错。通过推广快速最小逼近基插值技术，我们获得了码长下具有次二次复杂度的ILRS码（及其变种）高效译码方案。据我们所知，所提出的译码方案是首个能够在和秩度量和斜度量下纠正超出唯一译码区域差错的方案。通过蒙特卡洛仿真验证了所提译码方案的性能以及译码失败概率上界的紧致性。