On the Classification of Cyclic Hadamard Sequences(<Special Section>Sequence Design and its Application in Communications)

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Binary sequences with two-level periodic autocorrelation correspond directly to cyclic (ν, k, λ)-designs. When ν=4t-1, k=2t-1 and λ=t-1, for some positive integer t, the sequence (or design) is called a cyclic Hadamard sequence (or design). For all known examples, ν is either a prime number, a product of twin primes, or one less than a power of 2. Except when ν=2^k-1, all known examples are based on quadratic residues (using the Legendre symbol when ν is prime, and the Jacobi symbol when ν=p(p+2) where both p and p+2 are prime); or sextic residues (when ν is a prime of the form 4a^2+27). However, when ν=2^k-1, many constructions are now known, including m-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.
一般社団法人電子情報通信学会の論文
2006-09-01