NP-Hardness of Bounded Distance Decoding for Reed-Solomon Codes

Abstract: 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 $c&gt;0$ 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 $d=\lfloor n<sup>α\rfloor$ 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 $2^{{Θ(n^η\log} n)}=2<sup>{o(n)}$. 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 $q\ge d<sup>{2+ρ}$, for any fixed $ρ&gt;0$. 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.