双变量线性算子码
Bivariate Linear Operator Codes
AI总结：
本文提出了双变量线性算子（B-LO）码，推广了线性算子码框架以涵盖更多达到列表译码容量的码，并证明了其可列表译码的充分条件，统一了包括置换乘积码在内的多种已知容量可达码的译码结果。
摘要：
在这项工作中，我们提出了线性算子码族的一种推广，它能够涵盖更多达到列表译码容量的码。线性算子（LO）码由Bhandari、Harsha、Kumar和Sudan [BHKS24]引入，作为捕获容量可达码的一种方式。在他们的框架中，一个码由一组线性算子指定，这些算子被应用于消息多项式，然后在一组指定的求值点上进行求值。我们以一种可应用于双变量消息多项式的方式推广了这一思想，得到了我们所说的双变量线性算子（B-LO）码。我们展示了双变量线性算子码能够涵盖更多的容量可达码，包括由Berman、Shany和Tamo [BST24]引入的置换乘积码。这些码基于双变量消息多项式工作，这就是为什么我们的推广对于将它们作为线性算子框架的一部分来捕获是必要的。与关于线性算子码的初始论文类似，我们提出了双变量线性算子码可列表译码的充分条件。利用这一刻画，我们能够推导出将LO码的列表可译码性作为我们的B-LO码定理的一个特例的定理。我们还应用该定理来证明置换乘积码在达到容量前是可列表译码的，从而将这一结果与已知可列表译码的LO码的结果统一起来，包括折叠Reed-Solomon码、多重性码和仿射折叠Reed-Solomon码。
英文摘要：
In this work, we present a generalization of the linear operator family of codes that captures more codes that achieve list decoding capacity. Linear operator (LO) codes were introduced by Bhandari, Harsha, Kumar, and Sudan [BHKS24] as a way to capture capacity-achieving codes. In their framework, a code is specified by a collection of linear operators that are applied to a message polynomial and then evaluated at a specified set of evaluation points. We generalize this idea in a way that can be applied to bivariate message polynomials, getting what we call bivariate linear operator (B-LO) codes. We show that bivariate linear operator codes capture more capacity-achieving codes, including permuted product codes introduced by Berman, Shany, and Tamo [BST24]. These codes work with bivariate message polynomials, which is why our generalization is necessary to capture them as a part of the linear operator framework. Similarly to the initial paper on linear operator codes, we present sufficient conditions for a bivariate linear operator code to be list decodable. Using this characterization, we are able to derive the theorem characterizing list-decodability of LO codes as a specific case of our theorem for B-LO codes. We also apply this theorem to show that permuted product codes are list decodable up to capacity, thereby unifying this result with those of known list-decodable LO codes, including Folded Reed-Solomon, Multiplicity, and Affine Folded Reed-Solomon codes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
