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The authors extend the recent results on the list-decodability of randomly punctured Reed-Solomon (RS) codes up to the Singleton bound to the broader class of polynomial ideal (PI) codes.
They show that randomly punctured PI codes over an exponentially large alphabet exactly achieve the Singleton bound for list-decoding, while those over a polynomially large alphabet approximately achieve it.
To prove this, the authors develop two new technical ingredients: a polynomial-ideal GM-MDS theorem and a duality theorem for PI codes. These results may be of independent interest.
By combining their list-decodability results with the efficient list-decoding algorithm for a large subclass of PI codes from prior work, the authors obtain the first family of codes that can be efficiently list-decoded up to the Singleton bound for list-decoding, for arbitrary list sizes.
This includes natural families of codes like folded Reed-Solomon and multiplicity codes (over random evaluation points), which have additional useful properties like a multiplication property.
The authors leave open the problem of extending their results to the setting of algebraic geometry (AG) codes, to obtain efficiently list-decodable codes up to the Singleton bound over constant-size alphabets.
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arxiv.org
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