In this work, we present an abstract framework for some algebraic error-correcting codes with the aim of capturing codes that are list-decodable to capacity, along with their decoding algorithm. In the polynomial ideal framework, a code is specified by some ideals in a polynomial ring, messages are polynomials and their encoding is the residue modulo the ideals. We present an alternate way of viewing this class of codes in terms of linear operators, and show that this alternate view makes their algorithmic list-decodability amenable to analysis. Our framework leads to a new class of codes that we call affine Folded Reed-Solomon codes (which are themselves a special case of the broader class we explore). These codes are common generalizations of the well-studied Folded Reed-Solomon codes and Multiplicity codes, while also capturing the less-studied Additive Folded Reed-Solomon codes as well as a large family of codes that were not previously known/studied. More significantly our framework also captures the algorithmic list-decodability of the constituent codes. Specifically, we present a unified view of the decoding algorithm for ideal theoretic codes and show that the decodability reduces to the analysis of the distance of some related codes. We show that good bounds on this distance lead to capacity-achieving performance of the underlying code, providing a unifying explanation of known capacity-achieving results. In the specific case of affine Folded Reed-Solomon codes, our framework shows that they are list-decodable up to capacity (for appropriate setting of the parameters), thereby unifying the previous results for Folded Reed-Solomon, Multiplicity and Additive Folded Reed-Solomon codes.

翻译：本文提出了一个抽象框架，用于描述某些代数纠错码，旨在捕捉能够进行列表译码至容量的码及其译码算法。在多项式理想框架下，码由多项式环中的某些理想指定，消息为多项式，其编码是模这些理想的余数。我们提出了一种通过线性算子来审视这类码的替代视角，并表明这一视角使得它们的算法列表可译性易于分析。我们的框架产生了一类新码，称为仿射折叠里德-所罗门码（这些码本身是我们所探索的更广泛码类的特例）。这些码是经过充分研究的折叠里德-所罗门码和重数码的共同推广，同时涵盖了较少研究的加性折叠里德-所罗门码以及大量此前未知/未研究的码族。更重要的是，我们的框架还捕捉了组成码的算法列表可译性。具体而言，我们提出了理想理论码译码算法的统一视角，并表明可译性简化为对某些相关码距离的分析。我们证明，对该距离的良好约束能使得底层码达到容量性能，从而提供了对已知容量达到结果的统一解释。在仿射折叠里德-所罗门码的特例中，我们的框架表明这些码在适当参数设置下可列表译码至容量，由此统一了折叠里德-所罗门码、重数码和加性折叠里德-所罗门码的先前结果。