Geodesic
Minimal number of generators of the lattice of subspaces of a finite-dimensional linear space
Zapiski Nauchnykh Seminarov POMI, Modules and algebraic groups, Tome 114 (1982), pp. 148-149

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It is proved that a minimal generating system of the lattice of all subspaces of a finite-dimensional vector space over a finite field of $q$ elements contains at most $\max(q+3)$ elements. This bound does not depend on the dimension of the space. 
@article{ZNSL_1982_114_a12,
     author = {A. A. Kravchenko},
     title = {Minimal number of generators of the lattice of subspaces of a finite-dimensional linear space},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {148--149},
     publisher = {mathdoc},
     volume = {114},
     year = {1982},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1982_114_a12/}
}
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A. A. Kravchenko. Minimal number of generators of the lattice of subspaces of a finite-dimensional linear space. Zapiski Nauchnykh Seminarov POMI, Modules and algebraic groups, Tome 114 (1982), pp. 148-149. http://geodesic.mathdoc.fr/item/ZNSL_1982_114_a12/