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Two families of two-dimensional orthogonal complete complementary codes (2D-OCCC) are presented They can be constructed, for example, over binary or complex symbols. The new 2D-OCCC retain most of the properties of the 1D-OCCC. The autocorrelation function is equal to zero for all nonzero shifts in both dimensions of any signature selected from the code. The cross-correlation for any pair of different signatures is equal to zero for all possible shifts in both dimensions.