Higher-Weight Dowling Lattices (HWDL in short) are special families of geometric lattices introduced by Dowling in connection with coding theory. The elements of HWDLs are the F_q-linear subspaces of (F_q)ⁿ having a basis of vectors with Hamming weight bounded from above, ordered by inclusion. These lattices were further studied, among others, by Bonin, Kung, and more recently by Ravagnani.

In this talk, we define and investigate structural properties of the q-analogues of HWDLs, which we call rank-metric lattices (RML in short). Their elements are the F_q^m – linear subspaces of (F_q^m)ⁿ having a basis of vectors with rank weight bounded from above, ordered by inclusion. We determine which RMLs are supersolvable, computing their characteristic polynomials. In the second part of the talk, we establish a connection between RMLs and the problem of distinguishing between inequivalent rank-metric codes.

The new results in this talk are joint work with A. Ravagnani