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

2026-09-24

AI Summary

This study addresses the long-standing lack of a precise characterization of the computational complexity of bounded distance decoding for Reed-Solomon codes, where prior NP-hardness results were confined to logarithmic gaps. To overcome this limitation, the authors construct reductions leveraging the moment subset sum problem, the uniform positive completion theorem over prime fields, and Deligne’s point-counting estimates. These techniques extend hard-code-length constructions to polynomial gaps, establishing complexity bounds for decoding radii below the covering radius n^α for any fixed rational α. The primary contribution is proving that this decoding problem is NP-complete under deterministic polynomial-time many-one reductions. This result transcends previous limitations and provides a tight characterization of the computational complexity of Reed-Solomon code decoding.

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 $c>0$ 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.

Problem

• Bounded Distance Decoding
• Reed-Solomon Codes
• NP-completeness
• Covering Radius
• Computational Complexity

Innovation

• Bounded Distance Decoding
• Reed-Solomon Codes
• NP-Completeness
• Moments Subset Sum
• Uniform Positive-Completion Theorem