We study information leakage in secure linear network coding schemes based on nested rank-metric codes. We show that the amount of information leaked to an adversary that observes a subset of network links is characterized by the conditional rank function of a representable $q$-polymatroid associated with the underlying rank-metric code pair. Building on this connection, we introduce the notions of $q$-polymatroid ports and $q$-access structures and describe their structural properties. Moreover, we extend Massey's correspondence between minimal codewords and minimal access sets to the rank-metric setting and prove a $q$-analogue of the Brickell--Davenport theorem.

翻译：本研究基于嵌套秩度量码探讨安全线性网络编码方案中的信息泄露问题。我们证明，当攻击者观测网络链路的子集时，其可获取的信息量可通过底层秩度量码对所关联的可表示$q$-多拟阵的条件秩函数进行刻画。基于此关联性，我们引入$q$-多拟阵端口与$q$-访问结构的概念，并描述其结构特性。此外，我们将Massey关于最小码字与最小访问集之间的对应关系推广至秩度量场景，并证明了Brickell--Davenport定理的$q$-模拟形式。