🤖 AI Summary
This work generalizes the GM-MDS theorem to broader code families. First, it establishes the theorem for arbitrary polynomial codes—whose generator matrix columns are evaluations of linearly independent polynomials at distinct points—as well as their duals. Second, it extends the result to algebraic codes, where generator matrix columns are sampled from points on an irreducible algebraic variety not contained in any origin-passing hyperplane, including their duals. The methodology integrates tools from algebraic geometry (structural properties of irreducible varieties), polynomial interpolation, zero-pattern analysis, and dual code construction. The core contribution is a unified framework certifying the higher-order MDS property for both polynomial and algebraic-variety-based codes, revealing a deep connection between this combinatorial property and underlying algebraic-geometric structure. This provides a critical theoretical foundation for constructing random punctured algebraic geometry codes over constant-size fields that approach list-decoding capacity, thereby advancing the design of efficient capacity-approaching codes.
📝 Abstract
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 [Brakensiek, Dhar, Gopi, Zhang; 2024], where it is proved that randomly punctured algebraic-geometric (AG) codes achieve list-decoding capacity over constant-sized fields.
