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 = Sigma(k is an element of 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 and the weight distribution sequence are.
relative difference family variable-weight optical orthogonal code graph decomposition