Rank-Metric Lattices

We introduce the class of rank-metric geometric lattices and initiate study their structural properties. Rank-metric can be seen as $q$-analogues higher-weight Dowling lattices, defined by himself in 1971. fully characterize supersolvable compute characteristic polynomials. then concentrate on small whose polynomial we cannot compute, provide a formula for them under polynomiality assumption Whitney numbers first kind. The proof relies computational results theory vector codes, which review this paper from perspective lattices. More precisely, notion lattice-rank weights code investigate properties combinatorial invariants distinguishers inequivalent codes.