In a recent breakthrough [BGM23, GZ23, AGL23], it was shown that Reed-Solomon codes, defined over random evaluation points, are list decodable with \emph{optimal} list size with high probability, i.e., they attain the \emph{Singleton bound for list decoding} [ST20, Rot22, GST22]. We extend this result to a large subclass of \emph{polynomial ideal codes}, which includes several well-studied families of error-correcting codes such as Reed-Solomon codes, folded Reed-Solomon codes, and multiplicity codes. Our results imply that a large subclass of polynomial ideal codes with random evaluation points over exponentially large fields achieve the Singleton bound for list-decoding exactly; while such codes over quadratically-sized fields approximately achieve it. Combining this with the {efficient} list-decoding algorithms for polynomial ideal codes of [BHKS21], our result implies as a corollary that a large subclass of polynomial ideal codes (over random evaluation points) is \emph{efficiently} list decodable with {optimal} list size. To the best of our knowledge, this gives the first family of codes that can be {efficiently} list decoded with {optimal} list size (for all list sizes), as well as the first family of {linear} codes of rate $R$ that can be {efficiently} list decoded up to a radius of $1 -R-\epsilon$ with list size that is {polynomial} (and even linear) in $1/\epsilon$. Moreover, the result applies to natural families of codes with algebraic structure such as folded Reed-Solomon or multiplicity codes (over random evaluation points). Our proof follows the general framework of [BGM23, GZ23], where the main new ingredients are a \emph{duality theorem} for polynomial ideal codes, as well as a new \emph{algebraic folded GM-MDS theorem} (extending the algebraic GM-MDS theorem of [YH19, Lov21]), which may be of independent interest.

翻译：在近期突破性工作[BGM23, GZ23, AGL23]中，研究表明定义在随机评估点上的Reed-Solomon码能以高概率实现列表译码的*最优*列表大小，即达到[ST20, Rot22, GST22]提出的*列表译码Singleton界*。我们将此结果推广至*多项式理想码*的一个大类，该类包含若干经过充分研究的纠错码族，如Reed-Solomon码、折叠Reed-Solomon码和重数码。我们的结果表明：定义在指数级大域上随机评估点的大类多项式理想码恰好达到列表译码的Singleton界；而定义在二次型域上的此类码则近似达到该界。结合[BHKS21]中多项式理想码的{高效}列表译码算法，我们的结果推论出：定义在随机评估点上的大类多项式理想码能以{最优}列表大小实现{高效}列表译码。据我们所知，这是首个能对所有列表大小实现{高效}列表译码且具有{最优}列表大小的码族，同时也是首个能以{多项式}（甚至线性）于$1/\epsilon$的列表大小在$1-R-\epsilon$半径内实现高效列表译码的{线性}速率$R$码族。此外，该结果适用于具有代数结构的自然码族（如定义在随机评估点上的折叠Reed-Solomon码或重数码）。我们的证明遵循[BGM23, GZ23]的总体框架，其中主要创新点包括多项式理想码的*对偶定理*以及新的*代数折叠GM-MDS定理*（推广了[YH19, Lov21]的代数GM-MDS定理），这些结果可能具有独立研究价值。