Abstract
<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>
Highlights
For two integers 0 ≤ k ≤ n we denote by Gq(n, k) the set of all k-dimensional subspaces in Fnq
The minimum distance of a constant dimension code (CDC) C is defined as dS(C) = min{dS(U, W ) : U, W ∈ C, U = W }
The maximum possible cardinality of an (n, M, d, k)q CDC is denoted by Aq(n, d; k)
Summary
Let Fq be the finite field with q elements, i.e., q is a prime power. For two integers 0 ≤ k ≤ n we denote by Gq(n, k) the set of all k-dimensional subspaces in Fnq. A subset C ⊆ Gq(n, k) is called a constant dimension code (CDC) and its elements are called codewords. With respect to recent improved constructions we mention e.g. Most of the contained improvements fit into a general framework of a combination of subcodes of a specific shape that we will present here. All constructions are based on an interplay between the subspace, the Hamming, and the rank metric distance. We give general upper bounds for the mentioned subcodes with special shapes. These codes are mainly used to describe and control the combination of different subcodes to a constant dimension code. For the contained subcodes the rank metric plays an important role for the construction, see Subsection 2.2. Based on the underlying general construction strategy sufficient conditions for additing further codewords are described in Subsection 2.3.
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