Generalized Linkage Construction for Constant-Dimension Codes

Heinlein, Daniel

doi:10.1109/tit.2020.3038272

Cited by 22 publications

(11 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…More precisely, in Section 4 we consider constructions of CDCs based on rank metric codes. The results therein provide not only a generalization of several recent results [4,17,18,20,35,43], but they also offer a more general point of view with respect to techniques that have been previously investigated in the literature, as for instance the so called linkage construction [16,39]. In particular, by using rank metric codes in different variants, we are able to obtain CDCs that either give improved lower bounds for many parameters, including A 2 (12, 4; 4), A q (12, 6; 6), A q (4k, 2k; 2k), k ≥ 4 even, A q (10, 4; 5), or whose size matches the best known lower bounds.…”

Section: Introductionsupporting

confidence: 75%

“…We remark that . −1 Rank-metric codes of constant rank with a lower bound on the minimum rank-distance have been studied in [15] and generalized in [20,35]. Here we restrict ourselves on subcodes contained in additive MRD codes.…”

Section: Constructions Based On Rank Metric Codesmentioning

confidence: 99%

“…−1 We remark that in Lemma 4.1 we can increase the dimension of the ambient space of the CDC C i if we further restrict the possible ranks of the elements in M j for 1 ≤ j < i. For details for the special case l = 2 see [20,Theorem 24], which e.g. allows to also treat the improved linkage construction from [23] in that framework.…”

Section: Constructions Based On Rank Metric Codesmentioning

confidence: 99%

See 2 more Smart Citations

Combining subspace codes

Cossidente¹,

Kurz²,

Marino³

et al. 2023

AMC

View full text Add to dashboard Cite

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

show abstract

Section: Introductionsupporting

confidence: 75%

Section: Constructions Based On Rank Metric Codesmentioning

confidence: 99%

Section: Constructions Based On Rank Metric Codesmentioning

confidence: 99%

See 1 more Smart Citation

Combining subspace codes

Cossidente¹,

Kurz²,

Marino³

et al. 2023

AMC

View full text Add to dashboard Cite

show abstract

“…Here the occurring rank distributions are given by 0 1 3 15 , 2 7 3 9 , and 1 1 2 4 3 11 . Rank-metric codes of constant rank with a lower bound on the minimum rank-distance have been studied in [10] and generalized in [17,33]. As rank metric codes with a given minimum rank distance and an upper bound on the occurring ranks pop up here, we propose the study of their sizes as an interesting open research problem.…”

Section: Preliminaries and Review Of Constructions From The Literaturementioning

confidence: 99%

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

Kurz¹

2023

AMC

View full text Add to dashboard Cite

<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>

show abstract

“…One main problem for the constant dimension subspace coding is to determine the maximal possible size A q (n, d, k) of such a code for given parameters n, d, k, q. We refer to papers [9,7,8,22,11,15,24,1,14,20,19,3,18] and the nice webpage [12] for latest constructions and references. The presently known best constant dimension subspace codes for n ≤ 19, q ≤ 9 are listed in the table in the webpage [12].…”

Section: Introductionmentioning

confidence: 99%

Parameter-controlled inserting constructions of constant dimension subspace codes

Lao¹,

Chen²,

Weng³

et al. 2020

Preprint

View full text Add to dashboard Cite

A basic problem in constant dimension subspace coding is to determine the maximal possible size A q (n, d, k) of a set of k-dimensional subspaces in F n q such that the subspace distance satisfies dis(U, V ) = 2k − 2 dim(U ∩ V ) ≥ d for any two different k-dimensional subspaces U and V in this set. In this paper we propose new parameter-controlled inserting constructions of constant dimension subspace codes. These inserting constructions are flexible because they are controlled by parameters. Several new better lower bounds which are better than all previously constructive lower bounds can be derived from our flexible inserting constructions. 141 new constant dimension subspace codes of distances 4, 6, 8 better than previously best known codes are constructed.

show abstract

Generalized Linkage Construction for Constant-Dimension Codes

Cited by 22 publications

References 18 publications

Combining subspace codes

Combining subspace codes

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

Parameter-controlled inserting constructions of constant dimension subspace codes

Contact Info

Product

Resources

About