Relative Difference Families With Variable Block Sizes and Their Related OOCs

Seven infinite classes of relative difference families with variable block sizes are presented explicitly. In particular, a balanced (gv,g,K,1)-DF with g=Σ_k∈K[(k²-k)/2] is explicitly given for: (i) K={3,4,5} and every v coprime to 6; (ii) K={3,4,6}, {3,5,6} or {3,4,5,6} and every v coprime to 30. As far as the authors are aware, these difference families can be viewed as the first explicit constructions of infinite classes of optimal variable-weight optical orthogonal codes with more than two weights. It is observed, however, that there are infinitely many values of v for which an optimal (v,W,1,Q) -OOC exists, whatever the set of weights W and the weight distribution sequence Q are.