Common Complements of Linear Subspaces and the Sparseness of MRD Codes

Motivated by applications to the theory of rank-metric codes, we study problem estimating number common complements a family subspaces over finite field in terms cardinality and its intersection structure. We derive upper lower bounds for this number, along with their asymptotic versions as size tends infinity. then use these describe general behavior respect sparseness density, showing that decisive property is whether or not spaces be complemented negligible size. By specializing our results matrix spaces, obtain maximum-rank-distance (MRD) codes rank metric. In particular, answer an open question coding theory, proving MRD are sparse all parameter sets grows, only very few exceptions. also investigate density columns infinity, obtaining new bound. Using properties Euler function from show bound improves on known most sets. conclude paper establishing structural codes.