### Citations

342 | Improved decoding of Reed-Solomon and algebraic-geometry codes
- Guruswami, Sudan
(Show Context)
Citation Context ...f errors. A fundamental problem in list decoding is to find the tradeoff among the information rate, decoding radius and the list size. List decoding of rank metric codes has been extensively studies =-=[8]-=-, [10], [14], [16], [18], [19], [20], [21]. However, even for Gabidulin codes which share several common properties and results paralleling with Reed-Solomon codes, people have not found an effective ... |

161 |
Theory of codes with maximum rank distance
- Gabidulin
- 1985
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Citation Context ...ubset of Fnqm and we denote by rank(x) to be the rank of the corresponding m× n matrix of x. Similar to block codes, one can also derive the Singleton bound as shown below. Lemma 1. (Singleton Bound) =-=[3]-=- Let C ⊆ Fnqm be a rank metric code with minimum rank distance d, then logq |C| ≤ min{n(m− d+ 1),m(n− d+ 1)}. Hence, logq |C| ≤ m(n− d+ 1) since n ≤ m. A rank metric code C achieving the above Singlet... |

91 |
Bilinear forms over a finite field, with applications to coding theory
- Delsarte
- 1978
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Citation Context ... field and the distance between two codewords is defined as the rank of their difference. The concept of rank metric was first introduced by Hua [13], and then considered in coding theory by Delsarte =-=[2]-=-. By adapting the idea of Reed-Solomon code, Gabidulin [4] gave a construction of a class of rank metric codes which is optimal and achieves the Singleton bound. The problem of uniquely decoding the G... |

14 |
Bounds on the minimum support weights,
- Helleseth, Kløve, et al.
- 1995
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Citation Context ... be bounded as shown below. Lemma 2. [5] For 0 ≤ r ≤ min{m,n}, one has qr(m+n−r) < |BR(0, r)| < K−1q qr(m+n−r), where Kq = ∏∞ j=1(1− q−j). It is easy to see that Kq is an increasing function of q. In =-=[12]-=-, it is shown that K2 ≈ 0.2887. Hence, we have K−1q ≤ K−12 < 4. Since rank metric defines a distance, we have the Hamming and Gilbert-Varsharmov like bounds as well. Lemma 3. (Hamming Bound) [6] Let C... |

11 | Packing and covering properties of rank metric codes
- Gadouleau, Yan
- 2008
(Show Context)
Citation Context ... = 0 u−1∏ i=0 (qn−qi)(qm−qi) qu−qi if u ≥ 1. Hence, |BR(0, r)| = 1 + r∑ u=1 u−1∏ i=0 (qn − qi)(qm − qi) qu − qi . Furthermore, the volume of a rank metric ball can be bounded as shown below. Lemma 2. =-=[5]-=- For 0 ≤ r ≤ min{m,n}, one has qr(m+n−r) < |BR(0, r)| < K−1q qr(m+n−r), where Kq = ∏∞ j=1(1− q−j). It is easy to see that Kq is an increasing function of q. In [12], it is shown that K2 ≈ 0.2887. Henc... |

9 |
A.: List-decoding of subspace codes and rank-metric codes up to singleton bound
- Mahdavifar, Vardy
- 2012
(Show Context)
Citation Context ...mental problem in list decoding is to find the tradeoff among the information rate, decoding radius and the list size. List decoding of rank metric codes has been extensively studies [8], [10], [14], =-=[16]-=-, [18], [19], [20], [21]. However, even for Gabidulin codes which share several common properties and results paralleling with Reed-Solomon codes, people have not found an effective list decoding algo... |

8 | Properties of codes with the rank metric
- Gadouleau, Yan
- 2006
(Show Context)
Citation Context .... In [12], it is shown that K2 ≈ 0.2887. Hence, we have K−1q ≤ K−12 < 4. Since rank metric defines a distance, we have the Hamming and Gilbert-Varsharmov like bounds as well. Lemma 3. (Hamming Bound) =-=[6]-=- Let C ⊆ Fnqm be a rank metric code with minimum rank distance d, then∣∣∣∣BR(0, d− 12 )∣∣∣∣ ≤ qmn|C| . A rank metric code C achieving the above Hamming bound is called a perfect code. However, unlike ... |

7 |
Complexity of decoding Gabidulin codes
- Gadouleau, Yan
- 2008
(Show Context)
Citation Context ...lgorithm) have been proposed, and then in [15] a method based on a Welch-Berlekamp key equation have been provided. These two methods are most efficient for high-rate and low-rate codes, respectively =-=[7]-=-. List decoding, was introduced by Elias and Wonzencraft independently, is a relax version of unique decoding for which decoder allows to output a list of possible codewords. List decoding gives a pos... |

7 | List decoding of lifted Gabidulin codes via the Plücker embedding
- Trautmann, Silberstein, et al.
- 2013
(Show Context)
Citation Context ...em in list decoding is to find the tradeoff among the information rate, decoding radius and the list size. List decoding of rank metric codes has been extensively studies [8], [10], [14], [16], [18], =-=[19]-=-, [20], [21]. However, even for Gabidulin codes which share several common properties and results paralleling with Reed-Solomon codes, people have not found an effective list decoding algorithm with d... |

6 |
Studies on rank distance codes
- Babu
- 1995
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Citation Context ...ode with minimum rank distance d, then∣∣∣∣BR(0, d− 12 )∣∣∣∣ ≤ qmn|C| . A rank metric code C achieving the above Hamming bound is called a perfect code. However, unlike classical codes, it is shown in =-=[1]-=- that there exist no perfect rank metric codes. There are both finite and asymptotic versions of the covering bound (i.e., the Gilbert-Varsharmov bound) for rank metric codes. In this paper, we only s... |

6 |
Error and erasure decoding of rank-codes with a modified Berlekamp-Massey algorithm
- Richter, Plass
- 2004
(Show Context)
Citation Context ...act, it has been solved several times, by adapting the different approaches for unique decoding ReedSolomon codes to the linearized setting, starting with Gabidulin’s original paper [4], and later in =-=[17]-=- a method based on the Berlekamp-Massey algorithm (or the extended Euclidean algorithm) have been proposed, and then in [15] a method based on a Welch-Berlekamp key equation have been provided. These ... |

4 | Bounds on list decoding of rank-metric codes
- Wachter-Zeh
- 2013
(Show Context)
Citation Context ...ecoding is to find the tradeoff among the information rate, decoding radius and the list size. List decoding of rank metric codes has been extensively studies [8], [10], [14], [16], [18], [19], [20], =-=[21]-=-. However, even for Gabidulin codes which share several common properties and results paralleling with Reed-Solomon codes, people have not found an effective list decoding algorithm with decoding radi... |

2 |
and C.Wang, Explicit rank-metric codes list-decodable with optimal redundancy, arXiv:1311.7084 [cs.IT
- Guruswami
- 2013
(Show Context)
Citation Context ...pore A*STAR SERC under Research Grant 1121720011. Corresponding author: Y. Ding (email: dingyang@shu.edu.cn). ar X iv :1 40 1. 26 93 v2s[ cs .IT ]s23sJa n 2 01 4 2subspace designs, Guruswami and Wong =-=[11]-=- presented a deterministic algorithm to decode the subcodes of the Gadibulin codes constructed in [8]. Mahdavifar and Vardy [16] showed that one can list decode folded-Gabidulin codes with rate R and ... |

1 |
Rank-metric codes and their applications, manuscript
- Gabidulin
(Show Context)
Citation Context ...s the rank of their difference. The concept of rank metric was first introduced by Hua [13], and then considered in coding theory by Delsarte [2]. By adapting the idea of Reed-Solomon code, Gabidulin =-=[4]-=- gave a construction of a class of rank metric codes which is optimal and achieves the Singleton bound. The problem of uniquely decoding the Gabidulin codes up to half of minimum distance has received... |

1 |
On the list-decodability of random linear codes, 42nd Annual
- Hastad, Kopparty
(Show Context)
Citation Context ... we have −m+ (1−R− )(R+ )n ≥ 1, then (2) shows that |BR(x, ρn) ∩ C| is at least Ω(exp(n)). This completes the proof. IV. RANDOM Fq -LINEAR RANK METRIC CODES Inspired by the work of Guruswami et.al.=-=[9]-=-, we consider the list decodability of random Fq-linear rank metric codes where the code is viewed as an Fq-linear subspace of Fnqm . We show that if n/m tends to a constant b as n→∞, almost all rando... |

1 |
A theorem on matrices over a fields and its applications, Chinese mathematical society
- Hua
(Show Context)
Citation Context ...nk metric code, each codeword is a matrix over a finite field and the distance between two codewords is defined as the rank of their difference. The concept of rank metric was first introduced by Hua =-=[13]-=-, and then considered in coding theory by Delsarte [2]. By adapting the idea of Reed-Solomon code, Gabidulin [4] gave a construction of a class of rank metric codes which is optimal and achieves the S... |

1 |
A Welch-Berlekamp like algorithm for decoding Gabidulin codes
- Kotter, Kschischang
(Show Context)
Citation Context ... fundamental problem in list decoding is to find the tradeoff among the information rate, decoding radius and the list size. List decoding of rank metric codes has been extensively studies [8], [10], =-=[14]-=-, [16], [18], [19], [20], [21]. However, even for Gabidulin codes which share several common properties and results paralleling with Reed-Solomon codes, people have not found an effective list decodin... |

1 |
Coding for errors and erasures in random network coding
- Loidreau
(Show Context)
Citation Context ...earized setting, starting with Gabidulin’s original paper [4], and later in [17] a method based on the Berlekamp-Massey algorithm (or the extended Euclidean algorithm) have been proposed, and then in =-=[15]-=- a method based on a Welch-Berlekamp key equation have been provided. These two methods are most efficient for high-rate and low-rate codes, respectively [7]. List decoding, was introduced by Elias an... |

1 |
On the geometry of balls in the Guasssmannian and list decoding of lifted Gabidulin codes, available at http://arxiv.org/pdf/1309.0403.pdf
- Silberstein, Trautmann
(Show Context)
Citation Context ... problem in list decoding is to find the tradeoff among the information rate, decoding radius and the list size. List decoding of rank metric codes has been extensively studies [8], [10], [14], [16], =-=[18]-=-, [19], [20], [21]. However, even for Gabidulin codes which share several common properties and results paralleling with Reed-Solomon codes, people have not found an effective list decoding algorithm ... |

1 |
Bound on list decoding Gabidulin codes, available at http://arxiv.org/pdf/1205.0345.pdf
- Wachter-zeh
(Show Context)
Citation Context ...list decoding is to find the tradeoff among the information rate, decoding radius and the list size. List decoding of rank metric codes has been extensively studies [8], [10], [14], [16], [18], [19], =-=[20]-=-, [21]. However, even for Gabidulin codes which share several common properties and results paralleling with Reed-Solomon codes, people have not found an effective list decoding algorithm with decodin... |