The merit factor of binary sequences related to difference sets

Long binary sequences related to cyclic difference sets are investigated. Among all known constructions of cyclic difference sets we show that only sequences constructed from Hadamard difference sets can have an asymptotic nonzero merit factor. Maximal length shift register sequences, Legendre, and twin-prime sequences are all constructed from Hadamard difference sets. We prove that the asymptotic merit factor of any maximal length shift register sequence is three. For twin-prime sequences it is shown that the best asymptotic merit factor is six. This value is obtained by shifting the twin-prime sequence one quarter of its length. It turns out that Legendre sequences and twin-prime sequences have similar behavior. Based on the Jacobi symbol we investigate Jacobi sequences. The best asymptotic merit factor is shown to be six. Through the introduction of product sequences, it is argued that the maximal merit factor among all sequences of length N is at least six when N is large. We also demonstrate that it is fairly easy to construct sequences of moderate composite length with merit factor close to six. Index Terms -Sequences, correlation, cyclic difference sets.