NOTES ON FINITE GEOMETRIES AND THE CONSTRUCTION OF PBIB DESIGNS——IV.SOME“ANZAHL”THEOREMS IN ORTHOGONAL GEOMETRY OVER FINITE FIELDS OF CHARACTERISTIC NOT 2

Let F_q be a field with q elements where q is a power of an odd prime.And letS be an n×n nonsingular symmetric matrix.The set of all n×n matrices T satisfyingTST=S forms the orthogonal group On(Fq,S).It is well known that congruent non-singular symmetric matrices define isomorphic orthogonal groups.Thus for even n wemay take