Generalized GM-MDS: Polynomial Codes Are Higher Order MDS
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3618260.3649637.pdf
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Author(s)
Brakensiek, Joshua
Dhar, Manik
Gopi, Sivakanth
Date Issued
June 10, 2024
Publisher
ACM|Proceedings of the 56th Annual ACM Symposium on Theory of Computing
Citation
Brakensiek, Joshua, Dhar, Manik and Gopi, Sivakanth. 2024. "Generalized GM-MDS: Polynomial Codes Are Higher Order MDS."
Version
Final published version
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.
Description
STOC ’24, June 24–28, 2024, Vancouver, BC, Canada
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1145/3618260.3649637
