Coset Construction for Subspace Codes

Heinlein, Daniel; Kurz, Sascha

doi:10.1109/tit.2017.2753822

Cited by 31 publications

(18 citation statements)

References 26 publications

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“…Partitioning it into subcodes with subspace distance d > 2a i − 2b i is a hard problem in general and was e.g. considered in the context of the coset construction for CDCs, see [24]. If we restrict ourselves to LMRD codes, then one can determine an analytical lower bound.…”

Section: Constructions Based On Rank Metric Codesmentioning

confidence: 99%

Combining subspace codes

Cossidente¹,

Kurz²,

Marino³

et al. 2023

AMC

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In the context of constant-dimension subspace codes, an important problem is to determine the largest possible size Aq(n, d; k) of codes whose codewords are k-subspaces of F n q with minimum subspace distance d. Here in order to obtain improved constructions, we investigate several approaches to combine subspace codes. This allow us to present improvements on the lower bounds for constant-dimension subspace codes for many parameters, including

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Section: Constructions Based On Rank Metric Codesmentioning

confidence: 99%

Combining subspace codes

Cossidente¹,

Kurz²,

Marino³

et al. 2023

AMC

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“…Instead of starting with an FDRM code C F and lifting it to a CDC C F one can also start from an (m, N, d, k) q CDC C and an MRD code M ⊆ F k×(n−m) q with minimum rank distance d/2. With this we can construct a CDC [21], where the underlying construction was named coset construction. In [40] the inequality…”

Section: Preliminaries and Review Of Constructions From The Literaturementioning

confidence: 99%

“…Partitioning it into subcodes with subspace distance d > 2a i − 2b i is a hard problem in general and was e.g. considered in the context of the coset construction for CDCs, see [21]. We have a closer look at this problem in Subsection 3.2.…”

Section: Similar As For Thementioning

confidence: 99%

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The interplay of different metrics for the construction of constant dimension codes

Kurz¹

2023

AMC

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<p style='text-indent:20px;'>A basic problem for constant dimension codes is to determine the maximum possible size <inline-formula><tex-math id="M1">\begin{document}$ A_q(n,d;k) $\end{document}</tex-math></inline-formula> of a set of <inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula>-dimensional subspaces in <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{F}_q^n $\end{document}</tex-math></inline-formula>, called codewords, such that the subspace distance satisfies <inline-formula><tex-math id="M4">\begin{document}$ d_S(U,W): = 2k-2\dim(U\cap W)\ge d $\end{document}</tex-math></inline-formula> for all pairs of different codewords <inline-formula><tex-math id="M5">\begin{document}$ U $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ W $\end{document}</tex-math></inline-formula>. Constant dimension codes have applications in e.g. random linear network coding, cryptography, and distributed storage. Bounds for <inline-formula><tex-math id="M7">\begin{document}$ A_q(n,d;k) $\end{document}</tex-math></inline-formula> are the topic of many recent research papers. Providing a general framework we survey many of the latest constructions and show the potential for further improvements. As examples we give improved constructions for the cases <inline-formula><tex-math id="M8">\begin{document}$ A_q(10,4;5) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ A_q(11,4;4) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ A_q(12,6;6) $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M11">\begin{document}$ A_q(15,4;4) $\end{document}</tex-math></inline-formula>. We also derive general upper bounds for subcodes arising in those constructions.</p>

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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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Coset Construction for Subspace Codes

Cited by 31 publications

References 26 publications

Combining subspace codes

Combining subspace codes

The interplay of different metrics for the construction of constant dimension codes

Maximal Ferrers Diagram Codes: Constructions and Genericity Considerations

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