Geometry of classical groups over finite fields and its applications

Zhe-xian Wan
1997 Discrete Mathematics  
This 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;
more » ... (10) Concluding remarks. Z. -x. Wan~Discrete Mathematics 174 (1997) [365][366][367][368][369][370][371][372][373][374][375][376][377][378][379][380][381] of 0:q (n) such that TEGLn(D:q) carries the subspace P to PT. We may propose the following problems: (i) What are the orbits of subspaces of Fq (n) under the action of GLn(U:q)? (ii) How many orbits are there? (iii) What are the lengths of the orbits? (iv) What is the number of subspaces in an orbit contained in a given subspace? The answers to these four problems are well known, they are: (i) Two subspaces belong to the same orbit if and only if their dimensions are equal. (ii) There are altogether n + 1 orbits. (iii) Denote the length of the orbit of m-dimensional subspaces (O<<.m<~n) by N(m, n), then N(m,n) = Hinn-m+l (qi 1) (iv) The number of k-dimensional subspaces contained in a given m-dimensional subspace (O<<.k <~m<~n) is N(k,m).
doi:10.1016/s0012-365x(96)00350-0 fatcat:auiznbmbkzcfxczhq223ja6dmm