Reed-Solomon码有界距离解码的NP难度
NP-Hardness of Bounded Distance Decoding for Reed-Solomon Codes
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中文总结 AI 辅助
本文证明对任意固定有理数0<α<1/2，Reed-Solomon码在低于覆盖半径d=⌊n^α⌋处的有界距离解码是NP完全的，通过矩子集和归约实现。
中文摘要 AI 辅助
对于[n,K] Reed-Solomon码，覆盖半径为n-K。Gandikota、Ghazi和Grigorescu证明了当解码半径低于覆盖半径d时，对于每个1≤d≤c log n/log log n（其中c>0为绝对常数），有界距离解码是确定性的NP难的。我们证明，对于每个固定的有理数0<α<1/2，在显式表示的有限扩域上，对于低于覆盖半径的加性间隙d=⌊n^α⌋，有界距离解码在确定性多项式时间多一归约下是NP完全的。困难码具有奇数块长度n，维数K=(n+1)/2-d，解码半径(n-1)/2，码率趋于1/2。字母表大小在求值集大小上是次指数的：对于仅依赖于α的固定0<η<1，其为2^{Θ(n^η log n)}=2^{o(n)}。证明通过n-1个非零域元素上的矩子集和问题，要求子集大小为(n-1)/2且d个指定矩。算术成分是在素数域F_q上的一致正补全定理，其中q≥d^{2+ρ}，对于任意固定ρ>0。更精确的形式来自基于Deligne定理的高维点计数估计；我们归约中使用的较弱形式通过加性特征正交性、单变量Weil界、2d阶矩恒等式和Newton恒等式更初等地证明。一个通用补全池、一个扩域商构造和一个确定性线性大小的同步幂凝聚器完成了归约。
英文摘要
For an $[n,K]$ Reed--Solomon code, the covering radius is $n-K$. Gandikota, Ghazi, and Grigorescu proved deterministic NP-hardness of bounded-distance decoding when the decoding radius is $d$ below the covering radius for every $1\le d\le c\log n/\log\log n$, where $c>0$ is an absolute constant. We prove that, for every fixed rational $0<α<1/2$, bounded-distance decoding is NP-complete under deterministic polynomial-time many-one reductions over explicitly represented finite extension fields for the additive gap $d=\lfloor n^α\rfloor$ below the covering radius. The hard codes have odd block length~$n$, dimension $K=(n+1)/2-d$, decoding radius $(n-1)/2$, and rate tending to $1/2$. The alphabet size is subexponential in the evaluation set size: for a fixed $0<η<1$ depending only on $α$, it is $2^{Θ(n^η\log n)}=2^{o(n)}$. The proof passes through moments subset sum on $n-1$ nonzero field elements, with required subset size $(n-1)/2$ and $d$ prescribed moments. The arithmetic ingredient is a uniform positive-completion theorem over prime fields $\mathbb{F}_q$ with $q\ge d^{2+ρ}$, for any fixed $ρ>0$. A sharper form follows from a higher-dimensional point-count estimate based on Deligne's theorem; the weaker form used in our reduction is proved more elementarily using additive-character orthogonality, the one-variable Weil bound, a moment identity of order $2d$, and Newton identities. A universal completion pool, an extension-field quotient construction, and a deterministic linear-size simultaneous power condenser complete the reduction.
