In the context of network coding, one or multiple sources of information attempt to transmit messages to various terminals through a network of intermediate nodes (multicast). In order to maximize the network throughput, the nodes are allowed to recombine the received packets before forwarding them towards the sinks.

In this talk, I will give an introduction to network coding and its methods. In particular, I will introduce rank-metric codes as a solution to the problem of error amplification in network transmissions.

The second part of the talk is devoted to mathematical aspects of the theory of error-correcting codes. I will compare rank-metric codes with the more classical family of Hamming-metric codes, showing strong divergences in their behaviour with respect to various combinatorial properties.

In the last part of the talk I will briefly describe new lines of research intersecting coding theory, combinatorics, and number theory.