A Survey on Optical Orthogonal Codes

family of (0,1) sequences with desired autocorrelation and cross-correlation properties providing asynchronous multi-access communications with easy synchronization and good performance in OCDMA communication networks [1,3]. In this section we review a few of the most important algorithms in generating OOCs. An optical orthogonal code) , , , (c a w n λ λ is a family C of (0,1) sequences of length n with constant Hamming weight w satisfying the following two properties: 1. Autocorrelation property. For any codeword 0 1 1 n x x x C − = ∈ x , the inequality 1 0 n i i a i x x τ λ − ⊕ = ≤ ∑ holds for any integer 0(mod) n τ ≡ / , and 2. Cross-correlation property. For any two distinct codewords , C ∈ x y , the inequality 1 0 n i i c i x y τ λ − ⊕ = ≤ ∑ holds for any integer τ , where the notation ⊕ denotes the modulo-n addition [2]. When a c λ λ λ = = , we denote the OOC by (, ,) n w λ for simplicity. The number of codewords is called the size of the optical orthogonal code. From a practical point of view, a code with a large size is required [3]. To find the best possible codes, we need to determine an upper bound on the size of an OOC with the given parameters. Let) , , , (c a w n λ λ Φ be the largest possible size of an) , , , (c a w n λ λ OOC. An OOC achieving this maximum size is said to be optimal. It is easily shown that if (1) (1) a w w n λ − > − then (, , ,) 0 a c n w λ λ Φ = and if 2 c w n λ > then (, , ,) 1 a c n w λ λ Φ ≤ [4]. Based on the Johnson bound for constant-weight error correcting codes [7], we have the following bound [2]: