Mart{\'\i}nez-Pe{\~n}as and Kschischang (IEEE Trans.\ Inf.\ Theory, 2019) proposed lifted linearized Reed--Solomon codes as suitable codes for error control in multishot network coding. We show how to construct and decode \ac{LILRS} codes. Compared to the construction by Mart{\'\i}nez-Pe{\~n}as--Kschischang, interleaving allows to increase the decoding region significantly and decreases the overhead due to the lifting (i.e., increases the code rate), at the cost of an increased packet size. We propose two decoding schemes for \ac{LILRS} that are both capable of correcting insertions and deletions beyond half the minimum distance of the code by either allowing a list or a small decoding failure probability. We propose a probabilistic unique {\LOlike} decoder for \ac{LILRS} codes and an efficient interpolation-based decoding scheme that can be either used as a list decoder (with exponential worst-case list size) or as a probabilistic unique decoder. We derive upper bounds on the decoding failure probability of the probabilistic-unique decoders which show that the decoding failure probability is very small for most channel realizations up to the maximal decoding radius. The tightness of the bounds is verified by Monte Carlo simulations.

翻译：Martínez-Peñas 和 Kschischang（《IEEE 信息论汇刊》，2019）提出了提升线性化里德-所罗门码作为适用于多跳网络编码中差错控制的编码。我们展示了如何构造和译码 LILRS 码。与 Martínez-Peñas–Kschischang 的构造相比，交错允许显著增大译码区域，并通过提升操作减少开销（即提高码率），但代价是增加了数据包大小。我们为 LILRS 码提出了两种译码方案，这两种方案都能通过允许列表或较低的译码失败概率，纠正超过码最小距离一半的插入和删除错误。我们为 LILRS 码提出了一种概率唯一型似然译码器，并设计了一种高效的基于插值的译码方案，该方案既可作为列表译码器（具有指数级最坏情况列表大小）使用，也可作为概率唯一译码器使用。我们推导了概率唯一译码器译码失败概率的上界，结果表明在最大译码半径内，对于大多数信道实现，译码失败概率非常小。通过蒙特卡洛仿真验证了这些界的紧致性。