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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 algorithms. In the polynomial ideal framework, a code is specified by some ideals in a polynomial ring, messages are polynomials and the encoding of a message polynomial is the collection of residues of that polynomial 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 Univariate Multiplicity codes as well as the less-studied Additive Folded Reed-Solomon codes, and lead to 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 a good bound on this distance leads to a 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 efficiently 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. 

The multiplicity Schwartz-Zippel lemma bounds the total multiplicity of zeroes of a multivariate polynomial on a product set. This lemma motivates the multiplicity codes of Kopparty, Saraf and Yekhanin [J. ACM, 2014], who showed how to use this lemma to construct high-rate locally-decodable codes. However, the algorithmic results about these codes crucially rely on the fact that the polynomials are evaluated on a vector space and not an arbitrary product set. In this work, we show how to decode multivariate multiplicity codes of large multiplicities in polynomial time over finite product sets (over fields of large characteristic and zero characteristic). Previously such decoding algorithms were not known even for a positive fraction of errors. In contrast, our work goes all the way to the distance of the code and in particular exceeds both the unique-decoding bound and the Johnson radius. For errors exceeding the Johnson radius, even combinatorial list-decodablity of these codes was not known. Our algorithm is an application of the classical polynomial method directly to the multivariate setting. In particular, we do not rely on a reduction from the multivariate to the univariate case as is typical of many of the existing results on decoding codes based on multivariate polynomials. However, a vanilla application of the polynomial method in the multivariate setting does not yield a polynomial upper bound on the list size. We obtain a polynomial bound on the list size by taking an alternative view of multivariate multiplicity codes. In this view, we glue all the partial derivatives of the same order together using a fresh set z of variables. We then apply the polynomial method by viewing this as a problem over the field F(z) of rational functions in z .