Summary
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On the Classification of Cyclic Hadamard Sequences Solomon W. GOLOMB Publication IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences Vol.E89-A No.9 pp.2247-2253 Publication Date: 2006/09/01 Online ISSN: 1745-1337 DOI: 10.1093/ietfec/e89-a.9.2247 Print ISSN: 0916-8508 Type of Manuscript: Special Section INVITED PAPER (Special Section on Sequence Design and its Application in Communications) Category: Keyword: Hadamard sequences, cyclic Hadamard difference sets, cyclic (v, k, λ) designs, m-sequences, Legendre sequences, GMW sequences, Welch-Gong transformation, hyperoval constructions, Full Text: PDF(646.5KB)>>
Summary: Binary sequences with two-level periodic autocorrelation correspond directly to cyclic (v, k, λ)-designs. When v = 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, v is either a prime number, a product of twin primes, or one less than a power of 2. Except when v = 2k-1, all known examples are based on quadratic residues (using the Legendre symbol when v is prime, and the Jacobi symbol when v = p(p+2) where both p and p+2 are prime); or sextic residues (when v is a prime of the form 4a2 + 27). However, when v = 2k-1, many constructions are now known, including m-sequences (corresponding to Singer difference sets), quadratic and sextic residue sequences (when 2k-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. | ||||||||||
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