Rank-Metric Lattices

Giuseppe Cotardo; Alberto Ravagnani

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arXiv · 2206.09284

Rank-Metric Lattices

Giuseppe Cotardo·Alberto Ravagnani

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Abstract

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 the smallest rank-metric lattice whose characteristic polynomial we cannot compute, and provide a formula for it under a polynomiality assumption on its 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.

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Giuseppe Cotardo, Alberto Ravagnani. 2022-06-18. Rank-Metric Lattices. https://arxiv.org/abs/2206.09284

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