M(\mathbf{c}) $ and denoted by $ {\rm wt_R}(\mathbf{c}) $. In this paper, a code is a subset of $ \mathbb{F}_{q^m}^n $. The term *rank-metric code* is used for codes with the rank metric.

Universal secure linear network coding {#subsec universal secure}
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We consider a network with several sources and several sinks as in [@ahlswede; @linearnetwork]. In this model, a given source wants to transmit $ k $ packets in $ \mathbb{F}_q^m $ to one or several sink nodes, and does so by encoding them into a vector, $ \mathbf{c} \in \mathbb{F}_{q^m}^n $, which can be seen as $ n $ packets in $ \mathbb{F}_q^m $ by (\[eq def matrix representation\]), being $ n $ the number of outgoing links from the source.

In *linear network coding*, as considered in [@ahlswede] and [@linearnetwork], the nodes in the network forward linear combinations of received packets (see [@Koetter2003 Definition 1]), achieving higher throughput than just storing and forwarding. This means that a given sink is assumed to receive the vector $$\mathbf{y} = \mathbf{c} A^T \in \mathbb{F}_{q^m}^N,$$ for some matrix $ A \in \mathbb{F}_q^{N \times n} $, called a transfer matrix.

Two of the *main problems* in linear network coding considered in the 