On the classification of cyclic Hadamard sequences
Binary sequences with two-level periodic autocorrelation correspond directly to cyclic (nu, k, lambda)-designs. When nu = 4t - 1, k = 2t - 1 and lambda = t - 1, for some positive integer t, the sequence (or design) is called a cyclic Hadamard sequence (or design). For all known examples, 0 is either a prime number, a product of twin primes, or one less than a power of 2. Except when nu = 2(k) - 1, all known examples are based on quadratic residues (using the Legendre symbol when v is prime, and the Jacobi symbol when nu = p(p+2) where both p and p + 2 are prime); or sextic residues (when nu is a prime of the form 4a(2) + 27). However, when nu = 2(k) - 1, many constructions are now known, including in-sequences (corresponding to Singer difference sets), quadratic and sextic residue sequences (when 2(k) - 1 is prime), GMW sequences and their generalizations (when k is composite), certain term-by-term sums of three and of five m-sequences and more general sums of trace terms. several constructions based on hyper-ovals in finite geometries (found by Segre, by Glynn, and by Maschietti), and the result of performing the Welch-Gong transformation on some of the foregoing.
hadamard sequences gmw sequences m-sequences welch-gong transformation cyclic hadantard difference sets cyclic (v,k,lambda) designs legendre sequences hyperoval constructions