On Deciding Deep Holes of Reed-Solomon Codes

Abstract

For generalized Reed-Solomon codes, it has been proved [6] 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∈Fq for Reed-Solomon [q, k] q , k < q 1/7−ǫ , cannot be a deep hole, whenever f (x) is a polynomial of degree k + d for 1 ≤ d < q 3/13−ǫ .

Complexity of Decoding Positive-Rate Primitive Reed–Solomon Codes

“…The difficulty is caused by the combinatorial complication of the subset D with no structures. In fact, there is a straightforward way to reduce the subset sum problem in D to the deep hole problem of a generalized Reed-Solomon code, which can then be reduced to the maximal likelihood decoding problem [3]. Note that the subset sum problem for D ⊆ F q is hard only if |D| is much smaller than q.…”

“…The maximal likelihood decoding problem of RS q [q, k] is considered to be hard, but the attempts to prove its NP-completeness have failed so far. The methods in [6] [3] can not be specialized to RS q [q, k] because we have lost the freedom to select D. The only known complexity result [4] in this direction says that the decoding of RS q [q, k] is at least as hard as the discrete logarithm in F * q h for h satisfying…”

Complexity of Decoding Positive-Rate Reed-Solomon Codes

“…The proof explores the combinatorial complication of the subset D, thus requires that n is at most polylogarithmic in q. In fact, there is a straightforward way to reduce the subset sum problem in D to the deep hole problem of a Reed-Solomon code, which can then be reduced to the maximum likelihood decoding problem [2]. Note that the subset sum problem for D ⊆ F q is hard only if |D| is much smaller than q.…”

On Determining Deep Holes of Generalized Reed-Solomon Codes

“…For a linear code, deep holes are defined to be vectors that are further away from codewords than all other vectors. The problem of deciding whether a received word is a deep hole for generalized ReedSolomon codes is proved to be co-NP-complete [9] [5]. For the extended Reed-Solomon codes RSq(Fq, k), a conjecture was made to classify deep holes in [5].…”

“…The problem of deciding whether a received word is a deep hole for generalized ReedSolomon codes is proved to be co-NP-complete [9] [5]. For the extended Reed-Solomon codes RSq(Fq, k), a conjecture was made to classify deep holes in [5]. Since then a lot of effort has been made to prove the conjecture, or its various forms.…”