On Deciding Deep Holes of Reed-Solomon Codes
Qi Cheng, Elizabeth Murray
TL;DR
The work analyzes deep holes in Reed-Solomon codes, first delivering a simpler co-NP-complete proof for generalized RS and then focusing on standard RS by reducing the problem to rational-point questions on absolutely irreducible hypersurfaces. It develops a hypersurface framework, proves a key absolutely irreducible leading-form, and leverages Cafure–Matera and Schmidt bounds to show that for $1<k<q^{1/7-\epsilon}$ and $1\le d<q^{3/13-\epsilon}$, a received word generated by a polynomial of degree $k+d$ is not a deep hole. A parallel, elementary NP-hardness result for ML decoding is obtained via a subset-sum reduction, highlighting the difficulty of decoding RS codes and motivating further classification of deep holes. The results bridge decoding complexity with algebraic geometry over finite fields and open avenues for precise deep-hole characterizations in standard RS codes.
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 $(f(α))_{α\in \F_q}$ for Reed-Solomon $[q,k]_q$, $k < q^{1/7 - ε}$, cannot be a deep hole, whenever $f(x)$ is a polynomial of degree $k+d$ for $1\leq d < q^{3/13 -ε}$.