Geometry of classical groups over finite fields and its applications

AbstractThis is a survey paper on the geometry of classical groups over finite fields and its applications, old and new. The contents of the paper are as follows: (1) The seed of our study; (2) The problems we are interested in; (3) History; (4) Recent results; (5) Application to association schemes and designs; (6) Application to authentication codes; (7) Application to projective codes; (8) Application to lattices generated by orbits of subspaces; (9) Application to representations of forms by forms; (10) Concluding remarks