Bounds and constructions for (v,W,2,Q)(v,W,2,Q)-OOCs

In 1996, Yang introduced variable-weight optical orthogonal code for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. Let W={w1,…,wr}W={w1,…,wr} be an ordering of a set of rr integers greater than 11, λλ be a positive integer (auto- and cross-correlation parameter  ), and Q=(q1,…,qr)Q=(q1,…,qr) be an rr-tuple (weight distribution sequence  ) of positive rational numbers whose sum is 11. A (v,W,λ,Q)(v,W,λ,Q) variable-weight optical orthogonal code ((v,W,λ,Q)(v,W,λ,Q)-OOC) is a collection of (0,1)(0,1) sequences with weights in WW, auto- and cross-correlation parameter λλ. Some work has been done on the construction of optimal (v,W,1,Q)(v,W,1,Q)-OOCs, while little is known on the construction of (v,W,λ,Q)(v,W,λ,Q)-OOCs with λ≥2λ≥2. It is well known that (v,W,λ,Q)(v,W,λ,Q)-OOCs with λ≥2λ≥2 have much bigger cardinality than those of (v,W,1,Q)(v,W,1,Q)-OOCs for the same v,W,Qv,W,Q. In this paper, a new upper bound on the number of codewords of (v,W,λ,Q)(v,W,λ,Q)-OOCs is given, and infinite classes of optimal (v,{3,4},2,Q)(v,{3,4},2,Q)-OOCs are constructed.