Randomly punctured Reed–Solomon codes achieve list-decoding capacity over linear-sized fields
arxiv(2023)
摘要
Reed–Solomon codes are a classic family of error-correcting codes consisting
of evaluations of low-degree polynomials over a finite field on some sequence
of distinct field elements. They are widely known for their optimal
unique-decoding capabilities, but their list-decoding capabilities are not
fully understood. Given the prevalence of Reed-Solomon codes, a fundamental
question in coding theory is determining if Reed–Solomon codes can optimally
achieve list-decoding capacity.
A recent breakthrough by Brakensiek, Gopi, and Makam, established that
Reed–Solomon codes are combinatorially list-decodable all the way to capacity.
However, their results hold for randomly-punctured Reed–Solomon codes over an
exponentially large field size 2^O(n), where n is the block length of the
code. A natural question is whether Reed–Solomon codes can still achieve
capacity over smaller fields. Recently, Guo and Zhang showed that Reed–Solomon
codes are list-decodable to capacity with field size O(n^2). We show that
Reed–Solomon codes are list-decodable to capacity with linear field size
O(n), which is optimal up to the constant factor. We also give evidence that
the ratio between the alphabet size q and code length n cannot be bounded
by an absolute constant. Our techniques also show that random linear codes are
list-decodable up to (the alphabet-independent) capacity with optimal list-size
O(1/ε) and near-optimal alphabet size 2^O(1/ε^2),
where ε is the gap to capacity. As far as we are aware,
list-decoding up to capacity with optimal list-size O(1/ε) was
previously not known to be achievable with any linear code over a constant
alphabet size (even non-constructively). Our proofs are based on the ideas of
Guo and Zhang, and we additionally exploit symmetries of reduced intersection
matrices.
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