Efficient List-decoding of Polynomial Ideal Codes with Optimal List Size

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.