On Deciding Deep Holes of Reed-Solomon Codes

TL;DR

This paper simplifies the proof that determining deep holes in generalized Reed-Solomon codes is co-NP-complete and extends the analysis to standard Reed-Solomon codes, providing bounds on when a received word cannot be a deep hole.

Contribution

It offers a simpler proof for the co-NP-completeness of deep hole decision and establishes new bounds for standard Reed-Solomon codes using algebraic geometry techniques.

Findings

Simplified proof of co-NP-completeness for generalized Reed-Solomon codes

Reduction of deep hole problem to rational points on algebraic varieties

Bound on polynomial degree for which received words are not deep holes

Abstract

For generalized Reed-Solomon codes, it has been proved \cite{GuruswamiVa05} that the problem of determining if a received word is a deep hole is co-NP-complete. The reduction relies on the fact that the evaluation set of the code can be exponential in the length of the code -- a property that practical codes do not usually possess. In this paper, we first presented a much simpler proof of the same result. We then consider the problem for standard Reed-Solomon codes, i.e. the evaluation set consists of all the nonzero elements in the field. We reduce the problem of identifying deep holes to deciding whether an absolutely irreducible hypersurface over a finite field contains a rational point whose coordinates are pairwise distinct and nonzero. By applying Schmidt and Cafure-Matera estimation of rational points on algebraic varieties, we prove that the received vector $(f(\alpha))_{\alpha \in \F_q}$ for Reed-Solomon $[q,k]_q$, $k < q^{1/7 - \epsilon}$, cannot be a deep hole, whenever $f(x)$ is a polynomial of degree $k+d$ for $1\leq d < q^{3/13 -\epsilon}$.