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
[(
k
²
-
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.