The Study Of A Partition Of N-dimensional Affine Space And Its Shadow

Let AG(n,Fq)denote the n-dimensional affine space,and let ?(k,n)be the set of all k-flats in AG(n,Fq).A no-empty family F(?)?(k,n)is called t-intersecting if dim(F1?F2)?t for all F1,F2?F(n?k?t).Suppose that W1,W2,…,Wm denote some t-intersecting families in ?((k,n),if any k-flat in ?(k,n)belongs and only belongs to only one t-intersecting family Wi,then we say the set of {W1,W2,…,Wm}is a partition of ?(k,n)in n-dimensional affine space.Suppose that F(?)(k,n),we define the shadow of F to consist of those(k-1)-flats contained in at least one member of F.In this paper,we give the constructions for partition and shadow of n-dimensional affine space AG(n,Fq)and obtain the following results.1.For n-t?2k+2 and k(?)2t,there is no partition of ?(k,n)consisted by t-intersecting families which are not intersect with each other in the n-dimensional affine space AG(n,Fq);2.Suppose that q is a prime power,v is a 0-flat in AG(n,Fq),(?)F is the shadow of F,dF(v)is the degree of v with respect to F,and d(?)F(?)is the degree of v with respect to(?)F.If x>k,dF(v)?qy-k[y-1k-1]q and d(?)F(v)?qx-k+1[x-1 k-2]q,hen we have|(?)F|?qy-k+1[k-1 y]q.