IMPROVED LIST DECODING OF FOLDED REED-SOLOMON AND MULTIPLICITY CODES

Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf, Mary Wootters

Research output: Contribution to journalArticlepeer-review

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 source of recent advances in coding theory: folded RS codes were the first known explicit construction of capacity-achieving list decodable codes [V. Guruswami and A. Rudra, IEEE Trans. Inform. Theory, 54 (2008), pp. 135-150], and multiplicity codes were the first construction of high-rate locally decodable codes [S. Kopparty, S. Saraf, and S. Yekhanin, J. ACM, 61 (2014), 28]. In this work, we show that folded RS codes and multiplicity codes are in fact better than 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 [V. Guruswami and C. Wang, IEEE Trans. Inform. Theory, 59 (2013), pp. 3257-3268], [Z. Dvir and S. Lovett, Proc. 44th STOC, ACM, 2012, 351-358]. 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 we use this, together with additional ideas, to show that multivariate multiplicity codes are locally list decodable up to their minimum distance. By known reductions, this gives, in turn, capacity-achieving locally list decodable codes with query complexity exp(õ((log N)5/6)). This improves on the tensor-based construction of [B. Hemenway, N. Ron-Zewi, and M. Wootters, SIAM J. Comput., 49 (2019), pp. 157-195], which gave capacity-achieving locally list decodable codes of query complexity NÕ(1/log log N), and is close to the best known query complexity of exp(Õ(√log N)) for high-rate locally (uniquely) decodable codes .

Original languageEnglish
Pages (from-to)794-840
Number of pages47
JournalSIAM Journal on Computing
Volume52
Issue number3
DOIs
StatePublished - 2023

Keywords

  • error-correcting codes
  • folded Reed-Solomon codes
  • list decoding
  • local decoding
  • multiplicity codes

All Science Journal Classification (ASJC) codes

  • Computer Science(all)
  • Mathematics(all)

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