A basic problem for constant dimension codes is to determine the maximum possible size $A_q(n,d;k)$ of a set of $k$-dimensional subspaces in $\mathbb{F}_q^n$, called codewords, such that the subspace distance satisfies $d_S(U,W):=2k-2\dim(U\cap W)\ge d$ for all pairs of different codewords $U$, $W$. Constant dimension codes have applications in e.g.\ random linear network coding, cryptography, and distributed storage. Bounds for $A_q(n,d;k)$ are the topic of many recent research papers. Providing a general framework we survey many of the latest constructions and show up the potential for further improvements. As examples we give improved constructions for the cases $A_q(10,4;5)$, $A_q(11,4;4)$, $A_q(12,6;6)$, and $A_q(15,4;4)$. We also derive general upper bounds for subcodes arising in those constructions.
翻译：恒定维度代码的一个基本问题是确定以$$_q(n,d;k)确定一套以$\mathbb{F ⁇ q ⁇ n$为代号的美元维度子空间的最大可能大小$_q(n,d;k)$(美元),这样子空间距离就能够满足$d_S(U,W):=2k-2\dim(U\cap W)\ge d$(美元),对于不同编码的所有一对,不同编码字的所有一对,均需2k-2\dim(U\cap W)\ge d$(美元)。 恒定维度代码在例如\ 随机线性网络编码、加密和分布式存储中应用。$A_q(n,d;k)$。最近的许多研究论文都以美元为主题。提供一个总框架,我们调查了许多最新的构造,并展示了进一步改进的可能性。我们举例说,为这些子构建过程改进了工程的工程结构,例如$_q(11,4;4)$A_q(11,4;4美元)$A_q;6美元和$A_q(15,4;4美元)。
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