Multilength Optical Orthogonal Codes: New Upper Bounds and Optimal Constructions

Let N={n₀, n₁, ... , n_k-1} be a set of positive integers and M= {m₀, m₁, ... , m_k-1} be a multiset of positive integers. By an (N, M, w,1; λ)-multilength optical orthogonal code (MLOOC), we mean an MLOOC of autocross correlation value and intracross correlation value one and intercross correlation value λ. The code contains m_i codewords of weight w and length n_i for 0 ≤ i ≤ k-1. The study of MLOOCs is motivated by an application in optical networks requiring multiple signaling rates and quality-of-services. In this paper, we study (N, M, w,1; λ)-MLOOCs with λ =2 (the least value among the nontrivial intercross correlations). Some new upper bounds on code size are derived under certain restrictions and a novel encoding approach is established. A number of series of new MLOOCs are then produced. These codes are of optimal sizes with respect to the new bounds.