AG Codes Achieve List Decoding Capacity over Constant-Sized Fields
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3618260.3649651.pdf
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Author(s)
Brakensiek, Joshua
Dhar, Manik
Gopi, Sivakanth
Zhang, Zihan
Date Issued
June 10, 2024
Publisher
ACM| STOC 2024: Proceedings of the 56th Annual ACM Symposium on Theory of Computing
Citation
Brakensiek, Joshua, Dhar, Manik, Gopi, Sivakanth and Zhang, Zihan. 2024. "AG Codes Achieve List Decoding Capacity over Constant-Sized Fields."
Version
Final published version
Abstract
The recently-emerging field of higher order MDS codes has sought
to unify a number of concepts in coding theory. Such areas captured
by higher order MDS codes include maximally recoverable (MR)
tensor codes, codes with optimal list-decoding guarantees, and
codes with constrained generator matrices (as in the GM-MDS
theorem).
By proving these equivalences, Brakensiek-Gopi-Makam showed
the existence of optimally list-decodable Reed-Solomon codes over
exponential sized fields. Building on this, recent breakthroughs
by Guo-Zhang and Alrabiah-Guruswami-Li have shown that randomly punctured Reed-Solomon codes achieve list-decoding capacity (which is a relaxation of optimal list-decodability) over linear
size fields. We extend these works by developing a formal theory
of relaxed higher order MDS codes. In particular, we show that
there are two inequivalent relaxations which we call lower and
upper relaxations. The lower relaxation is equivalent to relaxed
optimal list-decodable codes and the upper relaxation is equivalent
to relaxed MR tensor codes with a single parity check per column.
We then generalize the techniques of Guo-Zhang and AlrabiahGuruswami-Li to show that both these relaxations can be constructed over constant size fields by randomly puncturing suitable
algebraic-geometric codes. For this, we crucially use the generalized GM-MDS theorem for polynomial codes recently proved by
Brakensiek-Dhar-Gopi. We obtain the following corollaries from
our main result:
Randomly punctured algebraic-geometric codes of rate š
are listdecodable up to radius šæ
šæ+1
(1 ā š
ā š) with list size šæ over fields of
size exp(š(šæ/š)). In particular, they achieve list-decoding capacity
with list size š(1/š) and field size exp(š(1/š
2
)). Prior to this work,
AG codes were not even known to achieve list-decoding capacity.
Description
STOC ā24, June 24ā28, 2024, Vancouver, BC, Canada
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1145/3618260.3649651
