2015/08/04 by Н. А. Балонин, N. A. Balonin, D. Z. Djokovic +1
Computer Science · Engineering · Mathematics · #Algorithm #Binary Golay code #Block code #Coding theory and cryptography #Combinatorics #Complementary sequences #Complex Hadamard matrix #Conjecture #Finite Group Theory Research #Hadamard matrix #Hadamard three-lines theorem #Hadamard transform #Hadamard's maximal determinant problem #Hamming code #Mathematical analysis #Mathematics #Ternary Golay code #graph theory and CDMA systems #math.CO
paper · pdf · doi:10.15217/issn1684-8853.2015.5.2
published as Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2015, no. 5, pp. 2-17 · 28 pages
arxiv created 2015/08/04 · openalex publication_date 2015/11/01 · arxiv updated 2015/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Apart from the ordinary and the periodic Golay pairs, we define also the negaperiodic Golay pairs. (They occurred first, under a different name, in a paper of Ito.) If a Hadamard matrix is also a Toeplitz matrix, we show that it must be either cyclic or negacyclic. We investigate the construction of Hadamard (and weighing matrices) from two negacyclic blocks (2N-type). The Hadamard matrices of 2N-type are equivalent to negaperiodic Golay pairs. We show that the Turyn multiplication of Golay pairs extends to a more general multiplication: one can multiply Golay pairs of length g and negaperiodic Golay pairs of length v to obtain negaperiodic Golay pairs of length gv. We show that the Ito's conjecture about Hadamard matrices is equivalent to the conjecture that negaperiodic Golay pairs exist for all even lengths.