scholarly journals Johnson Type Bounds for Mixed Dimension Subspace Codes

10.37236/8188 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Thomas Honold ◽  
Michael Kiermaier ◽  
Sascha Kurz
Keyword(s):  
Network Coding ◽  
Upper Bounds ◽  
Linear Network ◽  
Subspace Codes ◽  
Linear Network Coding ◽  
Constant Dimension ◽  
Random Linear Network Coding ◽  
Johnson Bound ◽  
Constant Dimension Codes ◽  
Mixed Dimension

Subspace codes, i.e., sets of subspaces of $\mathbb{F}_q^v$, are applied in random linear network coding. Here we give improved upper bounds for their cardinalities based on the Johnson bound for constant dimension codes.

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

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Sascha Kurz
Keyword(s):  
General Framework ◽  
Basic Problem ◽  
Distributed Storage ◽  
Upper Bounds ◽  
Linear Network ◽  
Research Papers ◽  
Linear Network Coding ◽  
Constant Dimension ◽  
Constant Dimension Codes ◽  
Subspace Distance

<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>

scholarly journals Exact Decoding Probability Under Random Linear Network Coding

2011 ◽  
Vol 15 (1) ◽  
pp. 67-69 ◽  
Author(s):  
Oscar Trullols-Cruces ◽  
Jose M. Barcelo-Ordinas ◽  
Marco Fiore
Keyword(s):  
Network Coding ◽  
Linear Network ◽  
Linear Network Coding ◽  
Random Linear Network Coding
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