In this paper, we study a new class of two-dimensional codes, here called multi-level prime codes, with expanded code cardinality by relaxing the maximum cross-correlation function to any arbitrary positive integer n. Besides having asymptotically optimal cardinality and zero autocorrelation sidelobes, these multi-level prime codes can be partitioned into n levels, such that the cross-correlation function between any two code matrices in level l is at most l, where l ∈ [1, n]. 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 allows the trade-off between code cardinality and performance for meeting different system requirements, such as capacity and throughput.
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