Small embedding of an S3(2, 4, u) into an Sλ(2, 4, u+w)

Let H be a subgraph of a graph G. An H-design (U,C) of order u and index @m is embedded into a G-design (V,B) of order v and index @l if @[email protected][email protected], [email protected]?V and there is an injective mapping f:C->B such that B is a subgraph of f(B) for every [email protected]?C. The mapping f is called the embedding of (U,C) into (V,B). We determine, for every admissible value of u and @l, the minimum value of w (except 12 values of (u,@l)) such that every S"3(2,4,u) can be embedded into an S"@l(2,4,u+w). This result implies that we determine also the minimum value of w such that there exists an S"@l(2,4,u+w) which embeds an E"2(u,1), where E"2 is the graph with two parallel edges and without isolated vertices.