Basic algebraic and combinatorial properties of finite vector spaces in which individual vectors are allowed to have multiplicities larger than 1 are derived. An application in coding theory is illustrated by showing that multispace codes that are introduced here may be used in random linear network coding scenarios, and that they generalize standard subspace codes (defined in the set of all subspaces of Fqn) and extend them to an infinitely larger set of parameters. In particular, in contrast to subspace codes, multispace codes of arbitrarily large cardinality and minimum distance exist for any fixed n and q.

中文翻译：

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矢量多空间和多空间代码


有限向量空间的基本代数和组合性质，其中允许单个向量具有大于 1 的多重性。通过表明此处介绍的多空间代码可用于随机线性网络编码场景，并且它们泛化标准子空间代码（在 Fqn 的所有子空间集中定义）并将它们扩展到无限大的参数集，来说明编码理论中的应用。特别是，与子空间代码相反，任何固定的 n 和 q 都存在具有任意大基数和最小距离的多空间代码。