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## Iterative list-decoding of Gabidulin codes via Gröbner based interpolation (2014)

Venue: | In arXiv:1405.7152 [cs.IT |

Citations: | 2 - 2 self |

### Citations

812 |
Finite fields
- Lidl, Niederreiter
- 1997
(Show Context)
Citation Context ...ight 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. =-=[8]-=- 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 ([8]... |

256 | Coding for errors and erasures in random network coding
- Kötter, Kschischang
- 2008
(Show Context)
Citation Context ...ceived word with respect to the rank metric is obtained. I. INTRODUCTION Over the last decade there has been increased interest in Gabidulin codes, mainly because of their relevance to network coding =-=[5]-=-, [20]. Gabidulin codes are optimal rank-metric nonbinary codes over a field Fmq (where q is a prime power). They were first derived by Gabidulin in [3] and independently by Delsarte in [2]. These cod... |

161 |
Theory of codes with maximum rank distance
- Gabidulin
- 1985
(Show Context)
Citation Context ...ly because of their relevance to network coding [5], [20]. Gabidulin codes are optimal rank-metric nonbinary codes over a field Fmq (where q is a prime power). They were first derived by Gabidulin in =-=[3]-=- and independently by Delsarte in [2]. These codes can be seen as the q-analog of Reed-Solomon codes, using q-linearized polynomials instead of arbitrary polynomials. They are optimal in the sense tha... |

158 | A rank-metric approach to error control in random network coding,”
- Silva, Kschischang, et al.
- 2008
(Show Context)
Citation Context ...d word with respect to the rank metric is obtained. I. INTRODUCTION Over the last decade there has been increased interest in Gabidulin codes, mainly because of their relevance to network coding [5], =-=[20]-=-. Gabidulin codes are optimal rank-metric nonbinary codes over a field Fmq (where q is a prime power). They were first derived by Gabidulin in [3] and independently by Delsarte in [2]. These codes can... |

99 |
Maximum-rank array codes and their application to crisscross error correction
- Roth
- 1991
(Show Context)
Citation Context ...also achieve the Singleton bound with respect to the rank metric and are thus MRD codes. They are not only of interest in network coding but also in space-time coding [11], crisscoss error correction =-=[15]-=- and distributed storage [18]. The decoding of Gabidulin codes has obtained a fair amount of attention in the literature, starting with work on decoding inside the unique decoding radius in [3], [4] a... |

91 |
Bilinear forms over a finite field, with applications to coding theory
- Delsarte
- 1978
(Show Context)
Citation Context ...ork coding [5], [20]. Gabidulin codes are optimal rank-metric nonbinary codes over a field Fmq (where q is a prime power). They were first derived by Gabidulin in [3] and independently by Delsarte in =-=[2]-=-. These codes can be seen as the q-analog of Reed-Solomon codes, using q-linearized polynomials instead of arbitrary polynomials. They are optimal in the sense that they are not only MDS codes with re... |

41 | On metrics for error correction in network coding,”
- Silva, Kschischang
- 2009
(Show Context)
Citation Context ...he roots of D(x) form a vector space of degree t which is equal to the span of e1, . . . , en (for this note that ei = f(gi) − ri). This is why D(x) is also called the error span polynomial (cf. e.g. =-=[19]-=-). 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. C. Gröbner bases... |

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 =-=[13]-=-. 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 Lq(x,... |

32 |
Maximum rank distance codes as space time codes
- Lusina, Gabidulin, et al.
- 2003
(Show Context)
Citation Context ...spect to the Hamming metric, but also achieve the Singleton bound with respect to the rank metric and are thus MRD codes. They are not only of interest in network coding but also in space-time coding =-=[11]-=-, crisscoss error correction [15] and distributed storage [18]. The decoding of Gabidulin codes has obtained a fair amount of attention in the literature, starting with work on decoding inside the uni... |

25 |
A Welch-Berlekamp like algorithm for decoding Gabidulin codes. In: Coding and cryptography
- Loidreau
- 2006
(Show Context)
Citation Context ...storage [18]. The decoding of Gabidulin codes has obtained a fair amount of attention in the literature, starting with work on decoding inside the unique decoding radius in [3], [4] and more recently =-=[10]-=-, [14], [16], [17], [21]. Decoding beyond the unique decoding radius was investigated in e.g. [9], [5], [12], [24], [25]. Related work on list-decoding of lifted Gabidulin codes can be found in [22]. ... |

16 | Fast encoding and decoding of Gabidulin codes
- Kschischang, Silva
(Show Context)
Citation Context ...ng of Gabidulin codes has obtained a fair amount of attention in the literature, starting with work on decoding inside the unique decoding radius in [3], [4] and more recently [10], [14], [16], [17], =-=[21]-=-. Decoding beyond the unique decoding radius was investigated in e.g. [9], [5], [12], [24], [25]. Related work on list-decoding of lifted Gabidulin codes can be found in [22]. Using the close resembla... |

15 |
S.: Fast decoding of rank-codes with rank errors and column erasures
- Richter, Plass
- 2004
(Show Context)
Citation Context ...e [18]. The decoding of Gabidulin codes has obtained a fair amount of attention in the literature, starting with work on decoding inside the unique decoding radius in [3], [4] and more recently [10], =-=[14]-=-, [16], [17], [21]. Decoding beyond the unique decoding radius was investigated in e.g. [9], [5], [12], [24], [25]. Related work on list-decoding of lifted Gabidulin codes can be found in [22]. Using ... |

10 |
A fast matrix decoding algorithm for rank-error-correcting codes
- Gabidulin
(Show Context)
Citation Context ... [15] and distributed storage [18]. The decoding of Gabidulin codes has obtained a fair amount of attention in the literature, starting with work on decoding inside the unique decoding radius in [3], =-=[4]-=- and more recently [10], [14], [16], [17], [21]. Decoding beyond the unique decoding radius was investigated in e.g. [9], [5], [12], [24], [25]. Related work on list-decoding of lifted Gabidulin codes... |

9 |
A.: List-decoding of subspace codes and rank-metric codes up to singleton bound
- Mahdavifar, Vardy
- 2012
(Show Context)
Citation Context ...rting with work on decoding inside the unique decoding radius in [3], [4] and more recently [10], [14], [16], [17], [21]. Decoding beyond the unique decoding radius was investigated in e.g. [9], [5], =-=[12]-=-, [24], [25]. Related work on list-decoding of lifted Gabidulin codes can be found in [22]. Using the close resemblance between Reed-Solomon codes and Gabidulin codes, the paper [10] translates Gabidu... |

8 |
M.: A parametric approach to list decoding of Reed-Solomon codes using interpolation
- Ali, Kuijper
- 2011
(Show Context)
Citation Context ...erent underlying fields we denote the rank of a matrix X over Fq by rankq(X). For some vector (v1, . . . , vn) ∈ Fnqm we denote the k × n Moore matrix by Mk(v1, . . . , vn) := v1 v2 . . . vn v =-=[1]-=- 1 v [1] 2 . . . v [1] n . . . v [k−1] 1 v [k−1] 2 . . . v [k−1] n , where [i] := qi. A q-linearized polynomial over Fqm is defined to be of the form f(x) = n∑ i=0 aix [i] , ai ∈ Fqm , where n ... |

8 |
Decoding rank errors beyond the error correcting capability
- Loidreau
- 2006
(Show Context)
Citation Context ...ature, starting with work on decoding inside the unique decoding radius in [3], [4] and more recently [10], [14], [16], [17], [21]. Decoding beyond the unique decoding radius was investigated in e.g. =-=[9]-=-, [5], [12], [24], [25]. Related work on list-decoding of lifted Gabidulin codes can be found in [22]. Using the close resemblance between Reed-Solomon codes and Gabidulin codes, the paper [10] transl... |

8 |
M.: Skew-feedback shift-register synthesis and decoding interleaved Gabidulin codes
- Sidorenko, Jiang, et al.
- 2011
(Show Context)
Citation Context ...decoding of Gabidulin codes has obtained a fair amount of attention in the literature, starting with work on decoding inside the unique decoding radius in [3], [4] and more recently [10], [14], [16], =-=[17]-=-, [21]. Decoding beyond the unique decoding radius was investigated in e.g. [9], [5], [12], [24], [25]. Related work on list-decoding of lifted Gabidulin codes can be found in [22]. Using the close re... |

8 |
A.: Interpolation-based decoding of interleaved Gabidulin codes
- Wachter-Zeh, Zeh
- 2013
(Show Context)
Citation Context ...with work on decoding inside the unique decoding radius in [3], [4] and more recently [10], [14], [16], [17], [21]. Decoding beyond the unique decoding radius was investigated in e.g. [9], [5], [12], =-=[24]-=-, [25]. Related work on list-decoding of lifted Gabidulin codes can be found in [22]. Using the close resemblance between Reed-Solomon codes and Gabidulin codes, the paper [10] translates Gabidulin de... |

7 | List decoding of lifted Gabidulin codes via the Plücker embedding
- Trautmann, Silberstein, et al.
- 2013
(Show Context)
Citation Context ...y [10], [14], [16], [17], [21]. Decoding beyond the unique decoding radius was investigated in e.g. [9], [5], [12], [24], [25]. Related work on list-decoding of lifted Gabidulin codes can be found in =-=[22]-=-. Using the close resemblance between Reed-Solomon codes and Gabidulin codes, the paper [10] translates Gabidulin decoding into a set of polynomial interpolation conditions. Essentially, this setup is... |

6 | Minimal Gröbner bases and the predictable leading monomial property. Linear Algebra and its
- Kuijper, Schindelar
(Show Context)
Citation Context ... leading positions in the two rows. Thus, after n steps, Bn is a minimal Gröbner basis for the interpolation module M(r). Consequently, Bn has the socalled Predictable Leading Monomial Property, see =-=[6]-=- and [1]. As a result of this property, the parametrization used for a(x) and c(x) in the second part of the algorithm will then yield all closest codewords. For the sake of brevity we omit the detail... |

6 |
M.: Decoding interleaved gabidulin codes and multisequence linearized shift-register synthesis
- Sidorenko, Bossert
- 2010
(Show Context)
Citation Context .... The decoding of Gabidulin codes has obtained a fair amount of attention in the literature, starting with work on decoding inside the unique decoding radius in [3], [4] and more recently [10], [14], =-=[16]-=-, [17], [21]. Decoding beyond the unique decoding radius was investigated in e.g. [9], [5], [12], [24], [25]. Related work on list-decoding of lifted Gabidulin codes can be found in [22]. Using the cl... |

4 | B.: General linearized polynomial interpolation and its applications
- Xie, Yan, et al.
- 2011
(Show Context)
Citation Context ...ork on decoding inside the unique decoding radius in [3], [4] and more recently [10], [14], [16], [17], [21]. Decoding beyond the unique decoding radius was investigated in e.g. [9], [5], [12], [24], =-=[25]-=-. Related work on list-decoding of lifted Gabidulin codes can be found in [22]. Using the close resemblance between Reed-Solomon codes and Gabidulin codes, the paper [10] translates Gabidulin decoding... |

3 | List decoding Gabidulin codes via interpolation and the Euclidean algorithm
- Kuijper, Trautmann
- 2014
(Show Context)
Citation Context ... cn) ∈ C is a codeword and e = (e1, . . . , en) ∈ F n qm is the error vector. We now recall the polynomial interpolation setup from [10] via a more general formulation in the next theorem. Theorem 4 (=-=[7]-=-, [10]). Let f(x) ∈ Lq(x, qm), qdeg(f(x)) < k and ci = f(gi) for i = 1, . . . , n. Then dR(c, r) = t if and only if there exists a D(x) ∈ Lq(x, qm), such that qdeg(D(x)) = t and D(ri) = D(f(gi)) ∀i ∈ ... |

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 ...und with respect to the rank metric and are thus MRD codes. They are not only of interest in network coding but also in space-time coding [11], crisscoss error correction [15] and distributed storage =-=[18]-=-. The decoding of Gabidulin codes has obtained a fair amount of attention in the literature, starting with work on decoding inside the unique decoding radius in [3], [4] and more recently [10], [14], ... |

3 |
Decoding of block and convolutional codes in rank metric
- Wachter-Zeh
- 2013
(Show Context)
Citation Context ...sily verified that the above polynomial is qlinearized and that Λg,r(gi) = ri for i = 1, . . . , n. Note that, although not under the same name, the previous two polynomials were also defined in e.g. =-=[23]-=-. In the following we will use matrix composition, which is defined analogously to matrix multiplication:[ a(x) b(x) c(x) d(x) ] ◦ [ e(x) f(x) g(x) h(x) ] := [ a(e(x)) + b(g(x)) a(f(x)) + b(h(x)) c(e(... |