On Deciding Deep Holes of Reed-Solomon Codes [article]

Qi Cheng, Elizabeth Murray
2005 arXiv   pre-print
For generalized Reed-Solomon codes, it has been proved 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
more » ... 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∈_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≤ d < q^3/13 -ϵ.
arXiv:cs/0509065v1 fatcat:lxaciygbsncwhizrc4kphm6ogy