Rank-metric codes are widely seen in a variety of applications ranging from storing information in the cloud to public-key cryptosystems. For over thirty years, the only optimal rank-metric codes known were Gabidulin codes. This changed when Sheekey recently constructed a new family of optimal rank-metric codes (MRD codes) using objects from finite geometry called linear sets on a projective line.

 

In this talk, I will explain the interplay between rank-metric codes, subspaces of linear maps and linear sets. I will show how we could extend Sheekey's connection to provide a one-to-one correspondence between MRD codes for all admissable parameters and a certain class of linear sets in higher dimension.