Combinatorial constructions of optimal optical orthogonal codes with weight 4

A (v,k,ý) optical orthogonal code C is a family of (0,1) sequences of length v and weight k satisfying the following correlation properties: 1) ý _0ýtýv-1x _tx _t+iýý for any x=(x ₀, x ₁, ..., x _v-1)ýC and any integer iý0(mod v); 2) ý _0ýtýv-1x _ty _t+iýý for any x=(x ₀, x ₁, ..., x _v-1)ýC, y=(y ₀, y ₁, ..., y _v-1)ýC with xýy, and any integer i, where the subscripts are taken modulo v. A (v,k,ý) optical orthogonal code (OOC) with └(1/k) └(v-1/k-2) └(v-2/k-2) └··· └(v-ý/k-ý) ┘$: M ┘ ┘ ┘ codewords is said to be optimal. OOCs are essential for success of fiber-optic code-division multiple-access (CDMA) communication systems. The use of an optimal OOC enables the largest possible number of asynchronous users to transmit information efficiently and reliably. In this paper, various combinatorial constructions for optimal (v,4,1) OOCs, such as those via skew starters and Weil's theorem on character sums, are given for vý0 (mod 12). These improve the known existence results on optimal OOCs. In particular, it is shown that an optimal (v,4,1) OOC exists for any positive integer vý0 (mod 24).