In a recent breakthrough [BGM23, GZ23, AGL23], it was shown that randomly punctured Reed-Solomon codes 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 the family of polynomial ideal codes, a large class of error-correcting codes which includes several well-studied families of codes such as Reed-Solomon, folded Reed-Solomon, and multiplicity codes. More specifically, similarly to the Reed-Solomon setting, we show that randomly punctured polynomial ideal codes over an exponentially large alphabet exactly achieve the Singleton bound for list-decoding; while such codes over a polynomially large alphabet approximately achieve it. Combining our results with the efficient list-decoding algorithm for a large subclass of polynomial ideal codes of [BHKS21], 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 $1 -R-\epsilon$ with list size that is polynomial (and even linear) in $1/\epsilon$. Our 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, AGL23], but several new ingredients are needed. The main two new ingredients are a polynomial-ideal GM-MDS theorem (extending the algebraic GM-MDS theorem of [YH19, Lov21]), as well as a duality theorem for polynomial ideal codes, both of which may be of independent interest.

翻译：在最近的突破性成果[BGM23, GZ23, AGL23]中，研究表明随机穿孔的Reed-Solomon码以高概率具有最优列表大小的列表可译性，即它们达到了列表译码的Singleton界[ST20, Rot22, GST22]。我们将这一结果推广到多项式理想码族，这是一大类纠错码，包含多个被深入研究的码族，如Reed-Solomon码、折叠Reed-Solomon码和重数码。具体而言，与Reed-Solomon码的情形类似，我们证明在指数级大字母表上，随机穿孔的多项式理想码精确达到了列表译码的Singleton界；而在多项式级大字母表上，此类码近似达到该界。将我们的结果与[BHKS21]中针对多项式理想码一大子类的高效列表译码算法相结合，可以推论出：多项式理想码的一大子类（在随机取值点上）具有最优列表大小的高效列表可译性。据我们所知，这给出了第一个能够以最优列表大小（对所有列表大小）进行高效列表译码的码族，同时也是第一个码率为$R$、能够高效列表译码至$1 -R-\epsilon$半径、且列表大小在$1/\epsilon$上为多项式（甚至线性）的线性码族。我们的结果适用于具有代数结构的自然码族，如折叠Reed-Solomon码或重数码（在随机取值点上）。我们的证明遵循[BGM23, GZ23, AGL23]的一般框架，但需要若干新的要素。两个主要的新要素是多项式理想GM-MDS定理（推广了[YH19, Lov21]的代数GM-MDS定理）以及多项式理想码的对偶定理，这两者可能都具有独立的研究价值。