The GM-MDS theorem, conjectured by Dau-Song-Dong-Yuen and proved by Lovett and Yildiz-Hassibi, shows that the generator matrices of Reed-Solomon codes can attain every possible configuration of zeros for an MDS code. The recently emerging theory of higher order MDS codes has connected the GM-MDS theorem to other important properties of Reed-Solomon codes, including showing that Reed-Solomon codes can achieve list decoding capacity, even over fields of size linear in the message length. A few works have extended the GM-MDS theorem to other families of codes, including Gabidulin and skew polynomial codes. In this paper, we generalize all these previous results by showing that the GM-MDS theorem applies to any \emph{polynomial code}, i.e., a code where the columns of the generator matrix are obtained by evaluating linearly independent polynomials at different points. We also show that the GM-MDS theorem applies to dual codes of such polynomial codes, which is non-trivial since the dual of a polynomial code may not be a polynomial code. More generally, we show that GM-MDS theorem also holds for algebraic codes (and their duals) where columns of the generator matrix are chosen to be points on some irreducible variety which is not contained in a hyperplane through the origin. Our generalization has applications to constructing capacity-achieving list-decodable codes as shown in a follow-up work by Brakensiek-Dhar-Gopi-Zhang, where it is proved that randomly punctured algebraic-geometric (AG) codes achieve list-decoding capacity over constant-sized fields.

翻译：GM-MDS定理由Dau-Song-Dong-Yuen猜想，并由Lovett与Yildiz-Hassibi证明，它揭示了Reed-Solomon码的生成矩阵可以实现在MDS码中零配置的所有可能模式。近期兴起的高阶MDS码理论将GM-MDS定理与Reed-Solomon码的其他重要性质联系起来，包括证明Reed-Solomon码能够达到列表译码容量，甚至在域大小与消息长度呈线性关系时也是如此。已有若干工作将GM-MDS定理推广至其他码族，包括Gabidulin码和斜多项式码。本文通过证明GM-MDS定理适用于任意\emph{多项式码}（即生成矩阵的列通过在不同点上评估线性无关多项式得到的码）来统一上述所有先前结果。我们还证明了GM-MDS定理适用于此类多项式码的对偶码——这一结论并非平凡，因为多项式码的对偶码可能不再是多项式码。更一般地，我们证明了GM-MDS定理同样适用于代数码（及其对偶码），其中生成矩阵的列被选为某个不可约簇（不包含于通过原点的超平面中）上的点。我们的推广在Brakensiek-Dhar-Gopi-Zhang的后续工作中被用于构造达到容量的列表可译码，该工作证明了随机打孔的代数几何（AG）码能够在固定大小的域上实现列表译码容量。