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Interlacing properties of shift-register sequences with generator polynomials irreducible over GF -herein called elementary sequences--are analyzed. The most important elementary sequences are maximal-length sequences ( -sequences). If the period of an elementary sequence is not prime, the sequence can always be constructed by interlacing shorter elementary sequences of period , provided divides . It is proved that all interlaced elementary sequences are generated by one and the same irreducible polynomial. Some relationships between equal-length elementary sequences are derived, including some rather unexpected crosscorrelation properties. As an example of an application of the theory, a new time-division multiplex technique for generating high-speed -sequences is presented.