Combinatorial Constructions for Optical Orthogonal Codes

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) ∑ 0≤t≤v−1xtxt+i ≤ λ for any x = (x0, x1, . . . , xv−1) ∈ C and any integer i ̸≡ 0 (mod v); (2) ∑ 0≤t≤v−1xtyt+i ≤ λ for any x = (x0, x1, . . . , xv−1) ∈ C, y = (y0, y1, . . . , yv−1) ∈ C with x ̸= y, and any integer i, where the subscripts are taken modulo v. A (v, k, λ) optical orthogonal code with ⌊ k ⌊ − 1 k − 1 ⌊ − 2 k − 2 ⌊· · ·⌊ − λ k − λ ⌋⌋⌋⌋⌋ codewords is said to be optimal. Optical orthogonal codes are essential for success of fiberoptic code division multiple access communication systems. The use of an optimal optical orthogonal code 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) optical orthogonal codes, 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 optical orthogonal codes. In particular, it is shown that an optimal (v, 4, 1) optical orthogonal code exists for any positive integer v ≡ 0 (mod 24). Index Terms – Combinatorial construction; Cyclic t-difference packing; Incomplete difference matrix; Optical orthogonal code; Optimal; Skew starter; Weil’s theorem.