NP-Hardness of Reed-Solomon Decoding, and the Prouhet-Tarry-Escott Problem

TL;DR

This paper proves NP-hardness for decoding Reed-Solomon codes at error rates below the maximum likelihood radius, introduces the Moments Subset Sum problem, and links it to the Prouhet-Tarry-Escott problem, highlighting new computational complexity insights.

Contribution

It establishes the first NP-hardness results for Reed-Solomon decoding at smaller radii and connects this to a new number theory problem, advancing understanding of decoding complexity.

Findings

NP-hard to decode beyond N-K- c(log N)/(log log N) errors

NP-hard under quasipolynomial reductions for errors > N-K- c log N

Introduces the Moments Subset Sum problem and links it to Prouhet-Tarry-Escott problem

Abstract

Establishing the complexity of {\em Bounded Distance Decoding} for Reed-Solomon codes is a fundamental open problem in coding theory, explicitly asked by Guruswami and Vardy (IEEE Trans. Inf. Theory, 2005). 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 $K=O(N)$, we show that it is NP-hard to decode more than $ N-K- c\frac{\log N}{\log\log N}$ errors (with $c>0$ an absolute constant). Moreover, we show that the problem is NP-hard under quasipolynomial-time reductions for an error amount $> N-K- c\log{N}$ (with $c>0$ an absolute constant). These results follow from the NP-hardness of a generalization of the classical Subset Sum problem to higher moments, called {\em 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.