Multilength Optical Orthogonal Codes: New Upper Bounds and Optimal Constructions

Abstract

Let $N=\{n_{0}, n_{1}, \ldots , n_{k-1}\}$ be a set of positive integers and $M= \{m_{0}, m_{1}, \ldots , m_{k-1}\}$ be a multiset of positive integers. By an $(N, M, w,1; \lambda )$ -multilength optical orthogonal code (MLOOC), we mean an MLOOC of autocross correlation value and intracross correlation value one and intercross correlation value $\lambda $ . The code contains $m_{i}$ codewords of weight $w$ and length $n_{i}$ for $0\le i\le k-1$ . 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 $(N, M, w,1; \lambda )$ -MLOOCs with $\lambda =2$ (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.