Decoding of random network codes

Gabidulin, E.M.; Pilipchuk, Nina I.; Bossert, Martin

doi:10.1134/s0032946010040034

Cited by 18 publications

(10 citation statements)

References 5 publications

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“…In F 6 2 , choose the spread code C 1 = Orb(F 2 3 ). This is a (6,9,6, 3) 2 code, where the cardinality follows from Corollary 3.8. In Let M ∈ GL 7 (F 2 ) be the companion matrix of the minimal polynomial of α.…”

Section: A Linkage Constructionmentioning

confidence: 99%

“…In [5], Etzion and Silberstein present a construction of subspace codes with large distance and cardinality which is based on rank-metric codes as introduced and studied earlier by Gabidulin in [8]. Decoding of such rank-metric codes is investigated by Gabidulin et al in [9], and the results are further applied to subspace codes for various combinations of errors and erasures.…”

Section: Introductionmentioning

confidence: 99%

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Cyclic orbit codes and stabilizer subfields

Gluesing-Luerssen¹,

Morrison²,

Troha³

2015

Advances in Mathematics of Communications

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Cyclic orbit codes are constant dimension subspace codes that arise as the orbit of a cyclic subgroup of the general linear group acting on subspaces in the given ambient space. With the aid of the largest subfield over which the given subspace is a vector space, the cardinality of the orbit code can be determined, and estimates for its distance can be found. This subfield is closely related to the stabilizer of the generating subspace. Finally, with a linkage construction larger, and longer, constant dimension codes can be derived from cyclic orbit codes without compromising the distance.

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Section: A Linkage Constructionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Cyclic orbit codes and stabilizer subfields

Gluesing-Luerssen¹,

Morrison²,

Troha³

2015

Advances in Mathematics of Communications

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“…This approach has led to the area of subspace codes and specifically to intensive research efforts on constructions of subspace codes with large subspace distance [23,21,25,10,14,9,16,12,15,11,24,29,17]. Most of the research focuses on constant-dimension codes (CDC's), that is, codes where all subspaces have the same dimension.…”

Section: Introductionmentioning

confidence: 99%

Construction of subspace codes through linkage

Gluesing-Luerssen¹,

Troha²

2016

AMC

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A construction is presented that allows to produce subspace codes of long length using subspace codes of shorter length in combination with a rank metric code. The subspace distance of the resulting code, called linkage code, is as good as the minimum subspace distance of the constituent codes. As a special application, the construction of the best known partial spreads is reproduced. Finally, for a special case of linkage, a decoding algorithm is presented which amounts to decoding with respect to the smaller constituent codes and which can be parallelized.

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“…For random linear network coding, see [5] by Chou et al and [18] by Ho et al, the natural codingtheoretical objects are subspace codes. This observation by Koetter et al [19,30] has led to extensive research efforts for constructions and decoding of subspace codes [3,8,9,12,13,14,15,16,17,21,24,29,30,32,34].…”

Section: Introductionmentioning

confidence: 99%

Maximal Ferrers Diagram Codes: Constructions and Genericity Considerations

Antrobus

Gluesing-Luerssen

2019

IEEE Trans. Inform. Theory

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This paper investigates the construction of rank-metric codes with specified Ferrers diagram shapes. These codes play a role in the multilevel construction for subspace codes. A conjecture from 2009 provides an upper bound for the dimension of a rank-metric code with given specified Ferrers diagram shape and rank distance. While the conjecture in its generality is wide open, several cases have been established in the literature. This paper contributes further cases of Ferrers diagrams and ranks for which the conjecture holds true. In addition, the proportion of maximal Ferrers diagram codes within the space of all rank-metric codes with the same shape and dimension is investigated. Special attention is being paid to MRD codes. It is shown that for growing field size the limiting proportion depends highly on the Ferrers diagram. For instance, for [m × 2]-MRD codes with rank 2 this limiting proportion is close to 1/e. of MRD codes that are not equivalent to Gabidulin codes and not necessarily F q m -linear. Most notably, in [27] Sheekey presents a construction of MRD-codes that are not equivalent to Gabidulin codes. Further contributions have been made by de la Cruz et al. [6] and Trombetti/Zhou [33].A very straightforward construction of good subspace codes with the aid of rank-metric codes is the lifting construction [19]: to each matrix M in the given rank-metric code one associates the row space of the matrix (I | M ), where I is the identity matrix of suitable size. While this simple construction leads to subspace codes with good distance, it usually does not produce large codes. A remedy has been introduced by Etzion/Silberstein [9]: obviously a matrix of the form (I | M ) ∈ F m×(m+n) is in reduced row echelon form (RREF) with pivot indices 1, . . . , m. This observation has led to the multilevel construction, where for each level a rank-metric code in F m×n is used to construct a subspace code in F m+n with all representing m × (m + n)-matrices being in RREF with a fixed set of general pivot indices. For this to work out properly, the matrices in the given rank-metric code have to be supported by the Ferrers diagram associated with the list of pivot indices; see [9] and Remark 2.5 later in this paper. As a result, the multilevel construction leads to the task of constructing large Ferrers diagram codes with a given rank distance. In [9] the authors provide an upper bound for the dimension of a rank-metric code supported by a given Ferrers diagram F and with a given rank distance δ. In this paper, codes attaining this bound will be called maximal [F; δ]-codes. To this day, it is not clear whether maximal [F; δ]-codes exist for all pairs (F; δ) and all finite fields. Several cases have been settled by Etzion et al. [8,9] and Gorla/Ravagnani [16] and, more recently, by Liu et al. [20] and Zhang/Ge [35], but the general case remains widely open. In [2] Ballico studies the existence of maximal [F; δ]-codes over number fields.In this paper we survey some of these results and extend them to further classes of pairs...

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Decoding of random network codes

Cited by 18 publications

References 5 publications

Cyclic orbit codes and stabilizer subfields

Cyclic orbit codes and stabilizer subfields

Construction of subspace codes through linkage

Maximal Ferrers Diagram Codes: Constructions and Genericity Considerations

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