Construction of Constant Dimension Codes From Two Parallel Versions of Linkage Construction

He, Xianmang

doi:10.1109/lcomm.2020.3012488

Cited by 16 publications

(6 citation statements)

References 12 publications

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“…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: 74%

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: Introductionsupporting

confidence: 74%

Combining subspace codes

Cossidente¹,

Kurz²,

Marino³

et al. 2023

AMC

View full text Add to dashboard Cite

show abstract

“…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 [4,17,18,19,35,43], 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 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: 74%

Combining subspace codes

Cossidente,

Kurz,

Marino

et al. 2019

Preprint

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In the context of constant-dimension subspace codes, an important problem is to determine the largest possible size A q (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 A q (10, 4; 5), A q (12, 4; 4), A q (12, 6, 6) and A q (16, 4; 4).

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“…Several of them started from Lemma 2.10 and improved Lemma 2.11 and Corollary 2.12, see e.g. [14,16,30,33,34]. We will briefly discuss this possibility in Subsection 3.3.…”

Section: Similar As For Thementioning

confidence: 99%

“…As an example we consider the parameters l = 2, n 1 = 5, n 2 = 5, a 1 = 2, a 2 = 3, b 1 = 1, and b 2 = 2, i.e., we are aiming at a lower bound for A q (10, 4; 5). Lemma 2.8 and Corollary 2.9 give a (10, , 4, 5) q CDC C with (14) #C = q 20 + [ 5 2 ] q • q 10 − q 7 − q 6 + q 2 + q − 1 + 1, i.e., #C = 1178312 for q = 2. For our specific choice n = (n 1 , n 2 ) = (5, 5), ā = (a 1 , a 2 ) = (2, 3), and b = (b 1 , b 2 ) = (1, 2) Corollary 2.12 gives a (10, , 4, 5) q CDC D such that C ∩ D = ∅ and d S (C ∪ D) ≥ 4, where #D ≥ q 9 , i.e., #D ≥ 512 for q = 2.…”

Section: Improved Constructionsmentioning

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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Construction of Constant Dimension Codes From Two Parallel Versions of Linkage Construction

Cited by 16 publications

References 12 publications

Combining subspace codes

Combining subspace codes

Combining subspace codes

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

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