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Multilevel Prime Codes for Optical CDMA Systems

In this paper, we study a new class of two-dimensional codes, here called multilevel prime codes, with expanded code cardinality by relaxing the maximum cross-correlation function to any arbitrary positive integer. Besides having asymptotically optimal cardinality and zero autocorrelation sidelobes, these multilevel prime codes can be partitioned into a tree structure of multiple levels of subsets of code matrices. In each level, the number of subsets, the number of code matrices per subset, and the cross-correlation function of each subset are related to the level number. The performance of the new codes in an optical code-division multiple-access system with hard-limiting detection is analyzed. Our results show that the unique partition property of the multilevel prime codes supports a trade-off between code cardinality and performance for meeting different system requirements, such as user capacity and throughput.