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 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 @math for Reed-Solomon @math , @math , cannot be a deep hole, whenever @math is a polynomial of degree @math for @math .
The question on decodability of Reed-Solomon codes has attracted attention recently, due to recent discoveries on the relationship between decoding Reed-Solomon codes and some number theoretical problems. Allowing exponential alphabets, Guruswami and Vardy proved that the maximum likelihood decoding is NP-complete. They essentially showed that deciding deep holes is co-NP-complete. When the evaluation set is precisely the whole field or @math , an NP-completeness result is hard to obtain, Cheng and Wan @cite_1 managed to prove that decoding problem of Reed-Solomon codes at certain radius is at least as hard as the discrete logarithm problem over finite fields. In this paper, we wish to establish an additional connection between decoding of standard Reed-Solomon codes and a classical number-theoretic problem -- that of determining the number of rational points on an algebraic hypersurface.
@cite_1
1565759886 2130539706 2077712820 2577154367
For generalized Reed-Solomon codes, it has been proved [7] 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 present 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 Cafure-Matera estimation of rational points on algebraic varieties, we prove that the received vector (f(α))α∈Fpfor the Reed-Solomon [p - 1, k]p, k < p1 4-Ɛ, cannot be a deep hole, whenever f(x) is a polynomial of degree k + d for 1 ≤ d ≤ p3 13-Ɛ. For an error-correcting code and a distance bound, the list decoding problem is to compute all the codewords within a given distance to a received message. The bounded distance decoding problem is to find one codeword if there is at least one codeword within the given distance, or to output the empty set if there is not. Obviously the bounded distance decoding problem is not as hard as the list decoding problem. For a Reed-Solomon code [n, k] sup q , a simple counting argument shows that for any integer 0 0. We show that the discrete logarithm problem over F sub qh can be efficiently reduced by a randomized algorithm to the bounded distance decoding problem of the Reed-Solomon code [q, g - h] sub q with radius q - g. These results show that the decoding problems for the Reed-Solomon code are at least as hard as the discrete logarithm problem over finite fields. The main tools to obtain these results are an interesting connection between the problem of list-decoding of Reed-Solomon code and the problem of discrete logarithm over finite fields, and a generalization of Katz's theorem on representations of elements in an extension finite field by products of distinct linear factors. The complexity of decoding the standard Reed-Solomon code is a well known open problem in coding theory. The main problem is to compute the error distance of a received word. Using the Weil bound for character sum estimate, we show that the error distance can be determined precisely when the degree of the received word is small. As an application of our method, we give a significant improvement of the recent bound of Cheng-Murray on non-existence of deep holes (words with maximal error distance). Reed-Solomon codes have found many applications in practical storage systems, but were until recently considered unsuitable for distributed storage applications due to the widely-held belief that they have poor repair bandwidth. The work of Guruswami and Wootters (STOC'16) has shown that one can actually perform bandwidth-efficient linear repair with Reed-Solomon codes: When the codes are over the field F q t and the number of parities r ≥ qs, where (t − s) divides t, there exists a linear scheme that achieves a repair bandwidth of (n − 1)(t − s) logg q bits. We extend this result by showing the existence of such a linear repair scheme for every 1 ≤ s t and r = qs. Additionally, we improve the lower bound on the repair bandwidth for Reed-Solomon codes, also established in the work of Guruswami and Wootters.