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The existence of binary sequences with specific aperiodic autocorrelation and cross correlation properties is investigated. Relationships are determined among the size of a sequence set, the length of the sequences n, the maximum autocorrelation sidelobe magnitude , and the maximum cross correlation magnitude . The principal result is the proof of the existence of sequence sets characterized by certain combinations of , and . The proof makes use of a new lower bound to the expected size of sequence sets constructed according to an explicit "random coding" procedure. For large , the sequence set size is controlled primarily by the cross correlation constraint . Two consequences of the existence theorem are 1) a demonstration that large sequence sets exist for which the maximum autocorrelation sidelobe and cross correlation magnitudes vanish almost as fast as the inverse square root of the sequence length a new proof of the Gilbert bound of coding theory.