Np-hardness of reed–solomon decoding, and the prouhet–tarry–escott problem

Venkata Gandikota, Badih Ghazi, Elena Grigorescu

Research output: Contribution to journalArticlepeer-review

Abstract

Establishing the complexity of bounded distance decoding for Reed–Solomon codes is a fundamental open problem in coding theory, explicitly asked by Guruswami and Vardy [IEEE Trans. Inform. Theory, 51 (2005), pp. 2249–2256]. The problem is motivated by the large current gap between the regime when it is NP-hard and the regime when it is efficiently solvable (i.e., the Johnson radius). We show the first NP-hardness results for asymptotically smaller decoding radii than the maximum likelihood decoding radius of Guruswami and Vardy. Specifically, for Reed–Solomon codes of length N and dimension K = Θ(N), we show that it is NP-hard to decode more than N − K − clog log log N N errors (with c > 0 an absolute constant). Moreover, we show that the problem is NP-hard under quasi-polynomial-time reductions for an error amount > N − K − clog N (with c > 0 an absolute constant). An alternative natural reformulation of the bounded distance decoding problem for Reed–Solomon codes is as a polynomial reconstruction problem. In this view, our results show that it is NP-hard to decide whether there exists a degree K polynomial passing through K + clog log N log N points from a given set of points (a1, b1), (a2, b2) . . ., (aN, bN). Furthermore, it is NP-hard under quasi-polynomial-time reductions to decide whether there is a degree K polynomial passing through K+clog N many points. These results follow from the NP-hardness of a generalization of the classical subset sum problem to higher moments, called moments subset sum, which has been a known open problem, and which May be of independent interest. We further reveal a strong connection with the well-studied Prouhet–Tarry–Escott problem in number theory, which turns out to capture a main Barrier in extending our techniques. We believe the Prouhet–Tarry–Escott problem deserves further study in the theoretical computer science community.

Original languageEnglish (US)
Pages (from-to)1547-1584
Number of pages38
JournalSIAM Journal on Computing
Volume47
Issue number4
DOIs
StatePublished - 2018
Externally publishedYes

Keywords

  • Bounded distance decoding
  • Polynomial reconstruction
  • Prouhet–Tarry–Escott problem
  • Reed–Solomon decdoing

ASJC Scopus subject areas

  • Computer Science(all)
  • Mathematics(all)

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