In this paper,the concepts of affine singular symplectic spaces ASG(2v + l, IFq)and singular symplectic group ASp2v+l,v(IFq)over IFq are given,and some Anzahl theorems in ASG(2v + l, IFq)are obtained using actions of ASP2V+l,v(IFq)on ASG(2v+l, IFq), Moreover,we construct an associative scheme and an authentication code in view of singular affine symplectic spaces.

The forming of concept of bogus symplectic space is studied.

The essential property and developmental course of Euclidean space,symplectic space and bogus symplectic space are studied.

In this paper,we construct the concept of bogus symplectic space,discuss the motion of bogus symplectic space and discribe the property of the motion by showing the composition and manifestation form of the motion,the generation and decomposition of the matrix of motion.

The association schemes of a kind of 2-dimensional subspaces of pseudo-symplectic space F_q~((2v+1+l)) and its structure;

After illustrating the conception pseudo-symplectry ,the evolvement and extension of the conception symplectry and the formation of pseudo-symplectic space,the nature of pseudo-symplectic space and the inner relationship and differences between pseudo-symplectic space and symplectic space are reflected here.