NP-hardness over polynomial-size alphabets

Establish deterministic NP-hardness for Reed–Solomon bounded-distance decoding with an additive gap of order n^α below the covering radius, in the parameter regime considered by the paper, over polynomial-size alphabets rather than the superpolynomial-size alphabets constructed here.

Background

The main theorem proves deterministic NP-completeness for Reed–Solomon bounded-distance decoding with gap d = floor(nα), for every fixed rational 0 < α < 1/2, but the constructed extension fields have size 2{Theta(nη log n)} = 2{o(n)}. Although this alphabet is subexponential in the block length, it remains superpolynomial in n.

The paper explicitly identifies polynomial-size alphabets as an unresolved quantitative strengthening of its hardness result. Achieving it would improve the alphabet-size complexity while retaining the growing additive gap below the covering radius.

References

This bound is still superpolynomial in~$n$; NP-hardness over polynomial-size alphabets remains open in this parameter regime.

— NP-Hardness of Bounded Distance Decoding for Reed-Solomon Codes  (2609.29120 - Wan et al., 24 Sep 2026) in Section 1, immediately after equation (intro-alphabet-size) in the Introduction

The most coding-theoretically direct open problem is to recover polynomial-moment hardness over prime fields.

— NP-Hardness of Bounded Distance Decoding for Reed-Solomon Codes  (2609.29120 - Wan et al., 24 Sep 2026) in Section 7, subsection “An alternative soundness construction over prime fields”