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 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，该定理表明里德-所罗门码的生成矩阵能够实现MDS码所有可能的零点配置。近期发展的高阶MDS码理论将GM-MDS定理与里德-所罗门码的其他重要性质联系起来，包括证明里德-所罗门码即使在消息长度线性规模的有限域上也能达到列表解码容量。已有若干研究将GM-MDS定理推广至其他码族，包括Gabidulin码与斜多项式码。本文通过证明GM-MDS定理适用于任意多项式码（即其生成矩阵的列通过对线性无关多项式在不同点求值获得），推广了所有先前结果。我们还证明该定理适用于此类多项式码的对偶码，这一结论非平凡，因为多项式码的对偶码未必仍是多项式码。更一般地，我们证明GM-MDS定理同样适用于代数码（及其对偶码），其生成矩阵的列选自不包含于过原点的超平面内的不可约簇上的点。如Brakensiek-Dhar-Gopi-Zhang在后续工作中所示，我们的推广结果可用于构建达到容量的列表可解码码，其中证明了随机穿孔的代数几何码在常数规模域上能够实现列表解码容量。