Browsing by Author "Ravagnani, Alberto"
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- Rank-Metric LatticesCotardo, Giuseppe; Ravagnani, Alberto (Electronic Journal Of Combinatorics, 2023-01-13)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 polyno-miality 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 inves-tigate their properties as combinatorial invariants and as code distinguishers for inequivalent codes.Mathematics Subject Classifications: 05A15, 06C10, 94B99