On the list and bounded distance decodability of the Reed-Solomon codes
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| Title: | On the list and bounded distance decodability of the Reed-Solomon codes |
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| Authors: | Qi Cheng, Daqing Wan |
| Contributors: | The Pennsylvania State University CiteSeerX Archives |
| Source: | http://www.math.uci.edu/~dwan/code.pdf. |
| Publisher Information: | IEEE Computer Society |
| Publication Year: | 2004 |
| Collection: | CiteSeerX |
| Description: | 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]q, a simple counting argument shows that for any integer 0 < g < n, there exists at least one Hamming ball of radius n−g, which contains at least � � n g−k g /q many codewords. Let ˆg(n, k, q) be the smallest positive integer g such that � � n g−k g /q < 1. One knows that k ≤ ˆg(n, k, q) ≤ √ nk ≤ n. For the distance bound up to n − √ nk, it is well known that both the list and bounded distance decoding can be solved efficiently. For the distance bound between n − √ nk and n − ˆg(n, k, q), we do not know whether the Reed-Solomon code is list, or bounded distance decodable, nor do we know whether there are polynomially many codewords in all balls of the radius. It is generally believed that the answers to both questions are no. There are public key cryptosystems proposed recently, whose security is based on the assumptions. In this paper, we prove: (1) List decoding can not be done for radius n − ˆg(n, k, q) or larger, otherwise the discrete logarithm over F q ˆg(n,k,q)−k is easy. (2) Let h and g be |
| Document Type: | text |
| File Description: | application/pdf |
| Language: | English |
| Relation: | http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.78.3972; http://www.math.uci.edu/~dwan/code.pdf |
| Availability: | http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.78.3972 http://www.math.uci.edu/~dwan/code.pdf |
| Rights: | Metadata may be used without restrictions as long as the oai identifier remains attached to it. |
| Accession Number: | edsbas.BB500C99 |
| Database: | BASE |
Nájsť tento článok vo Web of Science | Abstract: | 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]q, a simple counting argument shows that for any integer 0 < g < n, there exists at least one Hamming ball of radius n−g, which contains at least � � n g−k g /q many codewords. Let ˆg(n, k, q) be the smallest positive integer g such that � � n g−k g /q < 1. One knows that k ≤ ˆg(n, k, q) ≤ √ nk ≤ n. For the distance bound up to n − √ nk, it is well known that both the list and bounded distance decoding can be solved efficiently. For the distance bound between n − √ nk and n − ˆg(n, k, q), we do not know whether the Reed-Solomon code is list, or bounded distance decodable, nor do we know whether there are polynomially many codewords in all balls of the radius. It is generally believed that the answers to both questions are no. There are public key cryptosystems proposed recently, whose security is based on the assumptions. In this paper, we prove: (1) List decoding can not be done for radius n − ˆg(n, k, q) or larger, otherwise the discrete logarithm over F q ˆg(n,k,q)−k is easy. (2) Let h and g be |
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