Recursive constructions for difference matrices and relative difference families

We present a new recursive construction for difference matrices whose application allows us to improve some results by D. Jungnickel. For instance, we prove that for any Abelian pgroup G oftype $(n_1,n_2,...,n_t)$ there exists a $(G,p^e,1)$ difference matrix with $e =\lfloor{\sum_i n_i\over \max_i n_i}\rfloor$. Also, we prove that for any group G there exists a (G, p, 1) difference matrix where p is the smallest prime dividing |G|. Difference matrices are then used for constructing, recursively, relative difference families. We revisit some constructions by M. J. Colbourn, C. J. Colbourn, D. Jungnickel, K. T. Phelps, and R. M. Wilson. Combining them we get, in particular, the existence of a multiplier (G, k, λ)-DF for any Abelian group G of nonsquare-free order, whenever there exists a (p, k, λ)-DF for each prime p dividing |G|. Then we focus our attention on a recent construction by M. Jimbo. We improve this construction and prove, as a corollary, the existence of a (G, k, λ)-DF for any group G under the same conditions as above.