Higher-Order MDS Codes

Roth, Ron M.

doi:10.1109/tit.2022.3194521

Cited by 10 publications

(1 citation statement)

References 21 publications

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“…For Theorem 1.4, the connection between arbitrarily polynomial codes with the roots of polynomials is less clear. As such we generalize this technique of "repeated roots" within the higher order MDS framework of [BGM22,Rot22,BGM23]. Theorem 1.3 of [BGM23] shows that proving a GM-MDS theorem is equivalent to the following.…”

Section: Generalized Gm-mds Theorem For Monomial Codesmentioning

confidence: 99%

Lower Bounds for Maximally Recoverable Tensor Codes and Higher Order MDS Codes

Brakensiek

Gopi

Makam³

2022

IEEE Trans. Inform. Theory

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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 [BDGZ23], where it is proved that randomly punctured algebraic-geometric (AG) codes achieve list-decoding capacity over constant-sized fields.

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