Improved List Decoding of Folded Reed-Solomon and Multiplicity Codes

Abstract

We show new and improved list decoding properties of folded Reed-Solomon (RS) codes and multiplicity codes. Both of these families of codes are based on polynomials over finite fields, and both have been the sources of recent advances in coding theory: Folded RS codes were the first known explicit construction of capacity-achieving list decodable codes (Guruswami and Rudra, IEEE Trans. Information Theory , 2010), and multiplicity codes were the first construction of high-rate locally decodable codes (Kopparty, Saraf, and Yekhanin, J. ACM , 2014). In this work, we show that folded RS codes and multiplicity codes are in fact better than was previously known in the context of list decoding and local list decoding. Our first main result shows that folded RS codes achieve list decoding capacity with constant list sizes, independent of the block length. Prior work with constant list sizes first obtained list sizes that are polynomial in the block length, and relied on pre-encoding with subspace evasive sets to reduce the list sizes to a constant (Guruswami and Wang, IEEE Trans. Information Theory , 2012; Dvir and Lovett, STOC , 2012). The list size we obtain is (1 /ε ) O (1 /ε ) where ε is the gap to capacity, which matches the list size obtained by pre-encoding with subspace evasive sets. For our second main result, we observe that univariate multiplicity codes exhibit similar behavior, and use this, together with additional ideas, to show that multivariate multiplicity codes are locally list decodable up to their minimum distance . By known reduc-tions, this gives in turn capacity-achieving locally list decodable codes with query complexity exp( ˜ O ((log N ) 5 / 6 )). This improves on the tensor-based construction of (Hemenway, Ron-Zewi, and Wootters, SICOMP , 2019), which gave capacity-achieving locally list decodable codes of query complexity