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 optimal list size with high probability, i.e., they attain the Singleton bound for list decoding [ST20, Rot22, GST22]. We extend this result to a large subclass of 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 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-ϵ with list size that is polynomial (and even linear) in 1/ϵ. 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 duality theorem for polynomial ideal codes, as well as a new algebraic folded GM-MDS theorem (extending the algebraic GM-MDS theorem of [YH19, Lov21]), which may be of independent interest.