Valued rank-metric codes
Yassine El Maazouz, Marvin Anas Hahn, Alessandro Neri, and Mima Stanojkovski
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Submission date: 08. Apr. 2021
MSC-Numbers: 05E14, 11T71, 94B05
Keywords and phrases: rank-metric codes, discretely valued fields, Bruhat-Tits buildings, skew algebras, Mustafin varieties, MRD codes
Link to arXiv: See the arXiv entry of this preprint.
In this paper, we study linear spaces of matrices defined over discretely valued fields and discuss their dimension and minimal rank drops over the associated residue fields. To this end, we take first steps into the theory of rank-metric codes over discrete valuation rings by means of skew algebras derived from Galois extensions of rings. Additionally, we model projectivizations of rank-metric codes via Mustafin varieties, which we then employ to give sufficient conditions for a decrease in the dimension.