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 in fact generalize standard subspace codes (defined in the set of all subspaces of $ \mathbb{F}_q^n $) 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 $.

翻译：本文推导了允许单个向量具有大于1的重数的有限向量空间的基本代数与组合性质。通过展示本文引入的多重空间码可用于随机线性网络编码场景，并证明它们实际上推广了标准子空间码（定义在$\mathbb{F}_q^n$的所有子空间集合上），将其扩展至无限大的参数集，从而说明了编码理论中的一个应用。特别地，与子空间码相比，对于任意固定的$n$和$q$，存在任意大基数与最小距离的多重空间码。