We introduce the class of rank-metric geometric lattices and initiate the study of their structural properties. Rank-metric lattices can be seen as the $q$-analogues of higher-weight Dowling lattices, defined by Dowling himself in 1971. We fully characterize the supersolvable rank-metric lattices and compute their characteristic polynomials. We then concentrate on small rank-metric lattices whose characteristic polynomial we cannot compute, and provide a formula for them under a polynomiality assumption on their Whitney numbers of the first kind. The proof relies on computational results and on the theory of vector rank-metric codes, which we review in this paper from the perspective of rank-metric lattices. More precisely, we introduce the notion of lattice-rank weights of a rank-metric code and investigate their properties as combinatorial invariants and as code distinguishers for inequivalent codes.
@article{10_37236_11373,
author = {Giuseppe Cotardo and Alberto Ravagnani},
title = {Rank-metric lattices},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {1},
doi = {10.37236/11373},
zbl = {1539.06018},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11373/}
}
TY - JOUR
AU - Giuseppe Cotardo
AU - Alberto Ravagnani
TI - Rank-metric lattices
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/11373/
DO - 10.37236/11373
ID - 10_37236_11373
ER -
%0 Journal Article
%A Giuseppe Cotardo
%A Alberto Ravagnani
%T Rank-metric lattices
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/11373/
%R 10.37236/11373
%F 10_37236_11373
Giuseppe Cotardo; Alberto Ravagnani. Rank-metric lattices. The electronic journal of combinatorics, Tome 30 (2023) no. 1. doi: 10.37236/11373
