Aperiodic correlation constraints on large binary sequence sets

Abstract

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 \alpha , and the maximum cross correlation magnitude \beta . The principal result is the proof of the existence of sequence sets characterized by certain combinations of n, \alpha , and \beta . The proof makes use of a new lower bound to the expected size of sequence sets constructed according to an explicit random procedure. For large n , the sequence set size is controlled primarily by the cross correlation constraint \beta . 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 (l/\sqrt{n}); 2) a new proof of the Gilbert bound of coding theory.