Combinatorial constructions of optimal optical orthogonal codes with weight 4

A (v,k,λ) optical orthogonal code C is a family of (0,1) sequences of length v and weight k satisfying the following correlation properties: 1) Σ_0≤t≤v-1x_tx_t+i≤λ for any x=(x₀, x₁, ..., x_v-1)∈C and any integer i≠0(mod v); 2) Σ_0≤t≤v-1x_ty_t+i≤λ for any x=(x₀, x₁, ..., x_v-1)∈C, y=(y₀, y₁, ..., y_v-1)∈C with x≠y, and any integer i, where the subscripts are taken modulo v. A (v,k,λ) optical orthogonal code (OOC) with └(1/k)└(v-1/k-2)└(v-2/k-2)└···└(v-λ/k-λ)┘$: 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 v≡0 (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 v≡0 (mod 24).