From a (G,k,1) to a (C_k+G,k,1) difference family

Given a subgroup N of an additive group G, a (G,N,k,1) difference family (DF) is a set $\cal D$ of k-subsets of G such that $(d −d' | d, d' \in {\cal D}, d \neq d' , D \in {\cal D}) = G − N$. Generalizing a construction by Genma, Jimbo, and Mishima [4], we give a new condition for realizing a $(C_k \oplus G, C_k \times\{0\},k,1)-DF starting from a (G, {0}, k, 1)-DF. Among the consequences, new cyclic Steiner 2-designs are obtained.