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## List decoding Gabidulin codes via interpolation and the Euclidean algorithm (2014)

Venue: | IN ARXIV:1404.5716 [CS.IT |

Citations: | 3 - 3 self |

### Citations

812 |
Finite fields
- Lidl, Niederreiter
- 1997
(Show Context)
Citation Context ...with quotient m(x) if g(x) ◦m(x) = g(m(x)) = f(x). Efficient algorithms for all these operations (left and right symbolic multiplication and division) exist and can be found e.g. in [5]. Lemma 1 (cf. =-=[7]-=- Thm. 3.50). Let f(x) ∈ Lq(x, qm) and Fqs be the smallest extension field of Fqm that contains all roots of f(x). Then the set of all roots of f(x) forms a Fq-linear vector space in Fqs . Lemma 2 ([7]... |

256 | Coding for errors and erasures in random network coding
- Kötter, Kschischang
- 2008
(Show Context)
Citation Context ...ing metric, but also achieve the Singleton bound with respect to the rank metric and are thus MRD codes. There has been a rising interest in the last decade due to their application in network coding =-=[5]-=-, [19]. Since then a lot of work has been done on how to decode these codes. The question of minimum distance decoding inside the unique decoding radius has been addressed e.g. in [3], [4], [9], [13],... |

161 |
Theory of codes with maximum rank distance
- Gabidulin
- 1985
(Show Context)
Citation Context ...ent fields of coding theory, e.g. in (random) linear network coding [19], space-time coding [10], crisscoss error correction [14] and distributed storage [17]. They were first derived by Gabidulin in =-=[3]-=- and independently by Delsarte in [2]. These codes can be seen as the q-analog of ReedSolomon codes, using q-linearized polynomials instead of arbitrary polynomials over the finite field Fq (where q i... |

158 | A rank-metric approach to error control in random network coding,”
- Silva, Kschischang, et al.
- 2008
(Show Context)
Citation Context ...ed word with respect to the rank metric. I. INTRODUCTION Gabidulin codes are a family of optimal rank-metric codes, useful in different fields of coding theory, e.g. in (random) linear network coding =-=[19]-=-, space-time coding [10], crisscoss error correction [14] and distributed storage [17]. They were first derived by Gabidulin in [3] and independently by Delsarte in [2]. These codes can be seen as the... |

99 |
Maximum-rank array codes and their application to crisscross error correction
- Roth
- 1991
(Show Context)
Citation Context ...Gabidulin codes are a family of optimal rank-metric codes, useful in different fields of coding theory, e.g. in (random) linear network coding [19], space-time coding [10], crisscoss error correction =-=[14]-=- and distributed storage [17]. They were first derived by Gabidulin in [3] and independently by Delsarte in [2]. These codes can be seen as the q-analog of ReedSolomon codes, using q-linearized polyno... |

91 |
Bilinear forms over a finite field, with applications to coding theory
- Delsarte
- 1978
(Show Context)
Citation Context ...(random) linear network coding [19], space-time coding [10], crisscoss error correction [14] and distributed storage [17]. They were first derived by Gabidulin in [3] and independently by Delsarte in =-=[2]-=-. These codes can be seen as the q-analog of ReedSolomon codes, using q-linearized polynomials instead of arbitrary polynomials over the finite field Fq (where q is a prime power). They are optimal in... |

41 | On metrics for error correction in network coding,”
- Silva, Kschischang
- 2009
(Show Context)
Citation Context ...4. The previous theorem states that the roots of D(x) form a vector space of degree t which is equal to the span of e1, . . . , en. This is why D(x) is also called the error span polynomial (cf. e.g. =-=[18]-=-). The analogy in the classical Hamming metric set-up is the error locator polynomial, whose roots indicate the locations of the errors, and whose degree equals the number of errors. The interpolation... |

37 |
On a special class of polynomials
- Ore
- 1933
(Show Context)
Citation Context ...s defined to be of the form f(x) = n∑ i=0 aix [i] , ai ∈ Fqm , where n is called the q-degree of f(x), assuming that an 6= 0, denoted by qdeg(f). This class of polynomials was first studied by Ore in =-=[12]-=-. One can easily check that f(x1 + x2) = f(x1) + f(x2) and f(λx1) = λf(x1) for any x1, x2 ∈ Fqm and λ ∈ Fq , hence the name linearized. The set of all q-linearized polynomials over Fqm is denoted by L... |

32 | Maximum rank distance codes as space time codes - Lusina, Gabidulin, et al. - 2003 |

25 |
A Welch-Berlekamp like algorithm for decoding Gabidulin codes. In: Coding and cryptography
- Loidreau
- 2006
(Show Context)
Citation Context ...coding [5], [19]. Since then a lot of work has been done on how to decode these codes. The question of minimum distance decoding inside the unique decoding radius has been addressed e.g. in [3], [4], =-=[9]-=-, [13], [15], [16], [20], whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], [11], [22], [23]. Related work on list-decoding lifted Gabi... |

16 | Fast encoding and decoding of Gabidulin codes
- Kschischang, Silva
(Show Context)
Citation Context ... then a lot of work has been done on how to decode these codes. The question of minimum distance decoding inside the unique decoding radius has been addressed e.g. in [3], [4], [9], [13], [15], [16], =-=[20]-=-, whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], [11], [22], [23]. Related work on list-decoding lifted Gabidulin codes can be found... |

15 |
S.: Fast decoding of rank-codes with rank errors and column erasures
- Richter, Plass
- 2004
(Show Context)
Citation Context ...g [5], [19]. Since then a lot of work has been done on how to decode these codes. The question of minimum distance decoding inside the unique decoding radius has been addressed e.g. in [3], [4], [9], =-=[13]-=-, [15], [16], [20], whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], [11], [22], [23]. Related work on list-decoding lifted Gabidulin ... |

10 |
A fast matrix decoding algorithm for rank-error-correcting codes
- Gabidulin
(Show Context)
Citation Context ...work coding [5], [19]. Since then a lot of work has been done on how to decode these codes. The question of minimum distance decoding inside the unique decoding radius has been addressed e.g. in [3], =-=[4]-=-, [9], [13], [15], [16], [20], whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], [11], [22], [23]. Related work on list-decoding lifted... |

9 |
A.: List-decoding of subspace codes and rank-metric codes up to singleton bound
- Mahdavifar, Vardy
- 2012
(Show Context)
Citation Context ...ue decoding radius has been addressed e.g. in [3], [4], [9], [13], [15], [16], [20], whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], =-=[11]-=-, [22], [23]. Related work on list-decoding lifted Gabidulin codes can be found in [21]. In this work we explore list decoding further and, in contrast to the Sudan-Guruswami approach of [11], [22], p... |

8 |
M.: A parametric approach to list decoding of Reed-Solomon codes using interpolation
- Ali, Kuijper
- 2011
(Show Context)
Citation Context ...this work we explore list decoding further and, in contrast to the Sudan-Guruswami approach of [11], [22], present a parametric approach analogous to the one for list decoding Reed-Solomon codes from =-=[1]-=-. In a similar way as [9] we use interpolation, however unlike [9] we perform list decoding rather than unique decoding. A difference between our paper and the papers [9], [23] is that our approach is... |

8 |
Decoding rank errors beyond the error correcting capability
- Loidreau
- 2006
(Show Context)
Citation Context ... unique decoding radius has been addressed e.g. in [3], [4], [9], [13], [15], [16], [20], whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. =-=[8]-=-, [11], [22], [23]. Related work on list-decoding lifted Gabidulin codes can be found in [21]. In this work we explore list decoding further and, in contrast to the Sudan-Guruswami approach of [11], [... |

8 |
M.: Skew-feedback shift-register synthesis and decoding interleaved Gabidulin codes
- Sidorenko, Jiang, et al.
- 2011
(Show Context)
Citation Context ... Since then a lot of work has been done on how to decode these codes. The question of minimum distance decoding inside the unique decoding radius has been addressed e.g. in [3], [4], [9], [13], [15], =-=[16]-=-, [20], whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], [11], [22], [23]. Related work on list-decoding lifted Gabidulin codes can be... |

8 |
A.: Interpolation-based decoding of interleaved Gabidulin codes
- Wachter-Zeh, Zeh
- 2013
(Show Context)
Citation Context ...oding radius has been addressed e.g. in [3], [4], [9], [13], [15], [16], [20], whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], [11], =-=[22]-=-, [23]. Related work on list-decoding lifted Gabidulin codes can be found in [21]. In this work we explore list decoding further and, in contrast to the Sudan-Guruswami approach of [11], [22], present... |

7 | List decoding of lifted Gabidulin codes via the Plücker embedding
- Trautmann, Silberstein, et al.
- 2013
(Show Context)
Citation Context ...ereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], [11], [22], [23]. Related work on list-decoding lifted Gabidulin codes can be found in =-=[21]-=-. In this work we explore list decoding further and, in contrast to the Sudan-Guruswami approach of [11], [22], present a parametric approach analogous to the one for list decoding Reed-Solomon codes ... |

6 | Minimal Gröbner bases and the predictable leading monomial property. Linear Algebra and its
- Kuijper, Schindelar
(Show Context)
Citation Context ...) requirement 2) in Theorem 11. In fact, it can be proven that G is a minimal Gröbner basis for the interpolation module and has the so-called Predictable Leading Monomial Property analogous to [1], =-=[6]-=-. As a result of this property, the elements in the two for-loops that fulfill the divisibility requirement correspond to codewords with rank distance t = j−ℓ2+k−1 from r. Due to space limitations we ... |

6 |
M.: Decoding interleaved gabidulin codes and multisequence linearized shift-register synthesis
- Sidorenko, Bossert
- 2010
(Show Context)
Citation Context ... [19]. Since then a lot of work has been done on how to decode these codes. The question of minimum distance decoding inside the unique decoding radius has been addressed e.g. in [3], [4], [9], [13], =-=[15]-=-, [16], [20], whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], [11], [22], [23]. Related work on list-decoding lifted Gabidulin codes ... |

4 | B.: General linearized polynomial interpolation and its applications
- Xie, Yan, et al.
- 2011
(Show Context)
Citation Context ...radius has been addressed e.g. in [3], [4], [9], [13], [15], [16], [20], whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], [11], [22], =-=[23]-=-. Related work on list-decoding lifted Gabidulin codes can be found in [21]. In this work we explore list decoding further and, in contrast to the Sudan-Guruswami approach of [11], [22], present a par... |

3 |
S.: Adversarial error resilience in distributed storage using MRD codes and MDS array codes. arXiv:1202.0800v1 [cs.IT
- Silberstein, Rawat, et al.
- 2012
(Show Context)
Citation Context ...of optimal rank-metric codes, useful in different fields of coding theory, e.g. in (random) linear network coding [19], space-time coding [10], crisscoss error correction [14] and distributed storage =-=[17]-=-. They were first derived by Gabidulin in [3] and independently by Delsarte in [2]. These codes can be seen as the q-analog of ReedSolomon codes, using q-linearized polynomials instead of arbitrary po... |