Reed-Solomon译码的NP难性和Prouhet-Tarry-Escott问题
NP-Hardness of Reed-Solomon Decoding, and the Prouhet-Tarry-Escott Problem
摘要：
建立Reed-Solomon码的{ \em有界距离解码}的复杂度是编码理论中的基本开放问题，由Guruswami和Vardy（IEEE Trans. Inf. Theory，2005）明确提出。该问题是由当其为NP困难时的状态与当其有效可解时的状态（即，约翰逊半径（Johnson radius）我们展示了第一个NP-硬度的结果渐近较小的解码半径比最大似然解码半径的Guruswami和Vardy。具体来说，对于长度$N$和维度$K=O(N)$的Reed-Solomon码，我们证明了解码多于$ N-K- c\frac{\log N}{\log\log N}$错误是NP-难的（其中$c>0$是绝对常数）。此外，我们表明，问题是NP-困难下的准多项式时间减少的错误量$> N-K- c\log{N}$（与$c>0$的绝对常数）。这些结果遵循从NP-硬度的推广的经典子集和问题的更高的时刻，称为{ \em时刻子集和}，这一直是一个已知的开放问题，这可能是独立的利益。我们进一步揭示了一个强有力的连接，充分研究的Prouhet-Tarry-Escott问题数论，这是捕捉一个主要的障碍，在扩展我们的技术。我们认为Prouhet-Tarry-Escott问题值得理论计算机科学界进一步研究。
英文摘要：
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.
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