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

Venkata Gandikota, Badih Ghazi, Elena Grigorescu

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 $K=\Theta(N)$, 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 quasi-polynomial-time reductions for an error amount $> N-K- c\log{N}$ (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 $K+ c\frac{\log N}{\log\log N}$ points from a given set of points $(a_1, b_1), (a_2, b_2)\ldots, (a_N, b_N)$. Furthermore, it is NP-hard under quasi-polynomial-time reductions to decide whether there is a degree $K$ polynomial passing through $K+c\log{N}$ 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.