  We determine the number of ${\mathbb{F}}_q$-rational points of hyperplane
sections of classical determinantal varieties defined by the vanishing of
minors of a fixed size of a generic matrix, and identify sections giving the
maximum number of ${\mathbb{F}}_q$-rational points. Further we consider similar
questions for sections by linear subvarieties of a fixed codimension in the
ambient projective space. This is closely related to the study of linear codes
associated to determinantal varieties, and the determination of their weight
distribution, minimum distance and generalized Hamming weights. The previously
known results about these are generalized and expanded significantly
