Families of vector spaces with r-wise L-intersections

In this paper, we extend the well-known Frankl-Ray-Chaudhuri-Wilson Theorem on vector spaces to r-wise L-intersecting families, in both modular version and non-modular version. We also provide an upper bound on L-intersecting families of vector spaces which improves the Frankl-Ray-Chaudhuri-Wilson Theorem.