Complex Golay sequences: structure and applications Original Research Article

Complex Golay sequences were introduced in 1992 to generalize constructions for Hadamard matrices using Golay sequences. (In the last section of this paper we describe some independent earlier work on quadriphase pairs–equivalent objects used in the setting of signal processing.) Since then we have constructed some new infinite classes of these sequences and learned some facts about their structure. In particular, if the length of complex Golay sequences is divisible by a prime p≡3 mod 4, then their Hall polynomials have a nontrivial factorization h(x)k(x), cxdh(x)k∗(x) as polynomials over GF(p2), where c=a+bi, a2+b2≡−1 mod p and k∗ is obtained from k by a natural involution acting on complex Laurent polynomials. We explain how these facts can be used to simplify the search for complex Golay sequences, and show how to construct a large variety of sets of four complex sequences with zero autocorrelation, suitable for the construction of various matrices such as Hadamard matrices, complex Hadamard matrices and signed group Hadamard matrices over the dihedral signed group.