Common Complements of Linear Subspaces and the Sparseness of MRD Codes

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