A recent work of Goyal, Harsha, Kumar and Shankar gave nearly linear time algorithms for the list decoding of Folded Reed-Solomon codes (FRS) and univariate multiplicity codes up to list decoding capacity in their natural setting of parameters. A curious aspect of this work was that unlike most list decoding algorithms for codes that also naturally extend to the problem of list recovery, the algorithm in the work of Goyal et al. seemed to be crucially tied to the problem of list decoding. In particular, it wasn't clear if their algorithm could be generalized to solve the problem of list recovery FRS and univariate multiplicity codes in near linear time. In this work, we address this question and design $\tilde{O}(n)$-time algorithms for list recovery of Folded Reed-Solomon codes and univariate Multiplicity codes up to capacity, where $n$ is the blocklength of the code. For our proof, we build upon the lattice based ideas crucially used by Goyal et al. with one additional technical ingredient - we show the construction of appropriately structured lattices over the univariate polynomial ring that \emph{capture} the list recovery problem for these codes.

翻译：Goyal、Harsha、Kumar和Shankar近期的工作为折叠里德-所罗门码（FRS）和单变量重数码在其自然参数设定下达到列表译码容量的近乎线性时间算法提供了解决方案。该研究的一个有趣之处在于，与大多数同样能自然扩展到列表恢复问题的编码列表译码算法不同，Goyal等人提出的算法似乎与列表译码问题紧密绑定。具体而言，尚不清楚其算法能否推广至在近线性时间内解决折叠里德-所罗门码和单变量重数码的列表恢复问题。本文针对这一问题展开研究，设计了时间复杂度为$\tilde{O}(n)$的算法，用于实现折叠里德-所罗门码和单变量重数码达到容量限的列表恢复，其中$n$为码块长度。在证明过程中，我们在Goyal等人核心采用的格基思想基础上，引入一项关键技术要素——我们展示了在单变量多项式环上构造具有适当结构的格，这些格能够\emph{捕捉}上述编码的列表恢复问题。