Q 美元-多孔虫体及其与品级法典的关系 ($q$-Polymatroids and Their Relation to Rank-Metric Codes)

It is well known that linear rank-metric codes give rise to $q$-polymatroids. Analogously to classical matroid theory one may ask whether a given $q$-polymatroid is representable by a rank-metric code. We provide a partial answer by presenting examples of $q$-matroids that are not representable by ${\mathbb F}_{q^m}$-linear rank-metric codes. We then go on and introduce deletion and contraction for $q$-polymatroids and show that they are mutually dual and that they correspond to puncturing and shortening of rank-metric codes. Finally, we introduce a closure operator along with the notion of flats and show that the generalized rank weights of a rank-metric code are fully determined by the flats of the associated $q$-polymatroid.

翻译：众所周知,线性分级代码会产生美元-多式机器人。与典型的甲状腺机器人理论类似,人们可能会问,某一给定的美元-多式机器人是否可以用一等级代码代表。我们给出了部分答案,举例来说,美元-多式机器人是无法被美元=mathbb F ⁇ q ⁇ m} 美元-线性分级代码所代表的。然后,我们开始对美元-多式机器人进行删除和收缩,并表明它们是双向的,它们与分级代码的穿刺和缩短相对应。最后,我们引入了关闭操作员,同时提出了公寓概念,并表明一个分级代码的通用等级权重完全由相关的美元-多式机器人的平板决定。