Aperiodic correlation constraints on large binary sequence sets

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 coding" 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.