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：在近期突破性工作[BGM23, GZ23, AGL23]中，研究表明随机穿刺的里德-所罗门码能以高概率实现最优列表大小的列表解码，即达到列表解码的Singleton界[ST20, Rot22, GST22]。我们将这一结果推广至多项式理想码族——这是一类包含多种经典编码（如里德-所罗门码、折叠里德-所罗门码及重数码）的纠错码。具体而言，与里德-所罗门情形类似，我们证明了：在指数级大小的字母表上，随机穿刺的多项式理想码精确达到列表解码的Singleton界；而在多项式级大小的字母表上，则近似达到该界。将我们的结果与[BHKS21]中针对多项式理想码大子类的高效列表解码算法相结合，可推论出：基于随机求值点的多项式理想码大子类能以最优列表大小实现高效列表解码。据我们所知，这是首个可对所有列表大小实现最优列表大小高效解码的编码族，同时也是首个码率为$R$、能以关于$1/\epsilon$的多项式（甚至线性）列表大小高效解码至半径$1-R-\epsilon$的线性编码族。该结果适用于具有代数结构的自然编码族，如折叠里德-所罗门码或重数码（基于随机求值点）。我们的证明遵循[BGM23, GZ23, AGL23]的总体框架，但需要引入若干新工具。其中两个核心新工具是：多项式理想GM-MDS定理（扩展自[YH19, Lov21]的代数GM-MDS定理）以及多项式理想码的对偶定理——这两者可能具有独立研究价值。