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 $ \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：本文推导了有限向量空间的基本代数与组合性质，其中允许单个向量具有大于$1$的重数。通过证明此处引入的多重空间编码可用于随机线性网络编码场景，阐明了其在编码理论中的应用，并表明该编码推广了标准子空间编码（定义于$ \mathbb{F}_q^n $的所有子空间集合），且将其参数集扩展至无穷大。特别地，与子空间编码不同，对于任意固定的$n$和$q$，均存在具有任意大基数与最小距离的多重空间编码。