Perfect difference families and related variable-weight optical orthogonal codes

Perfect (v,K, 1) difference families ((v,K, 1)-PDF in short, and (v, k, 1)-PDF when K = {k}) were introduced by Ge et al. for their useful application to the construction of properly centered permutations, which can be used to construct new radar arrays. Some works had been done on the existences of (v, k, 1)-PDFs, while little is known when |K| ≥ 2 except for partial results by Ge et al. In 1996, Yang introduced variable-weight optical orthogonal code (variableweight OOC in short) for multimedia optical CDMA systems with multiple quality of service requirements. Jiang et al. presented general constructions of optimal variable-weight OOCs via (v,K, 1)-PDFs. Suppose K is a set of of positive integers, s is a positive integer, and s 6∈ K, a (v, {K, s∗}, 1)-PDF is a (v,K ⋃ {s}, 1)-PDF that contains only one block of size s. In this paper, by using perfect Langford sequences, the existences of (v, {3, s∗}, 1)-PDFs are completely solved for 4 ≤ s ≤ 7. New optimal variable-weight OOCs are then obtained.