Some New Classes Of Optimal Variable-Weight Optical Orthogonal Codes

Let W ={w1,w2,…,wr} be an ordering of a set of r integers greater than 1, and let Q = {q1,q2, …,qr} be an r-tuple of positive rational num-bers whose sum is 1. A (v,W,1, Q) variable-weight optical orthogonal code C,briefly (v, W, 1, Q)-OOC, is a collection of binary v-tuples satisfying the follow-ing three properties: (1) Every v-tuple in C has a Hamming weight contained in W, and there are exactly qi|C| codewords of weight wi, 1≤ i ≤ r. (2) For any codeword x = (x0,x1,…,xv-1) ∈ C with Hamming weight wi and any integer a (?) 0 (mod v), ∑t=0v-1 xtxt+σ ≤ 1. (3) For x ≠ y, x =(x0, x1, …,xv-1)∈ C,y = (y0, y1,…,yv-1) ∈ C,and any integer σ, ∑t=0v-1 xtyt+σ ≤ 1,where all subscripts are taken modulo v.Variable-weight optical orthogonal code was first introduced by Yang in 1993 for multimedia optical CDMA systems with multi quality of service re-quirements. A lot of researchers have been involved in the research of the constructions for optimal OOCs since then. There are many known results on(v,W,1,Q)-OOCs for Q ∈ {{3/4,1/4},{1/2,2/3},{2/3,1/3},{1/2,1/2}}and W (?) {3,4,5}.In this thesis, several combinational constructions for (v,W, 1,Q)-OOCs are presented by using difference matrices, skew starters and Weil’s theorem on character sums. We show the existence of the following 5 classes of optimal(v,{3,4},1,Q)-OOCs: (1)Q = {3/4,1/4} and v=30u, (2) Q = {1/3,2/3}and prime v = 30t + 1, (3) Q = {3/4,1/4} and prime v = 30t + 1, (4)Q = {2/3,1/3}, v = 2u and u = 12t + 1 is a prime, and (5) Q = {2/3,1/3},v = 4u and u = 12t + 7 is a prime.