ТЕОРЕТИЧЕСКАЯ И ПРИКЛАДНАЯ МАТЕМАТИКА UDC 519.614 doi:10.15217/issn1684-8853.2015.5.2 NEGAPERIODIC GOLAY PAIRS AND HADAMARD MATRICES N. A. Balonina, Dr. Sc., Tech., Professor, korbendfs@mail.ru D. Z. Djokovicb, PhD, Distinguished Professor Eme ritus, djokovic@uwaterloo.ca aSaint-Petersburg State University of Aerospace Instrumentation, 67, B. Morskaia St., 190000, Saint-Petersburg, Russian Federation bUniversity of Waterloo, Department of Pure Mathematics and Institute for Quantum Computing, Waterloo, Ontario, N2L 3G1, Canada Purpose: In analogy with the ordinary and the periodic Golay pairs, we introduce also the negaperiodic Golay pairs. (They occurred first, under a different name, in a paper of Ito.) Methods: We investigate the construction of Hadamard (and weighing) matrices from two negacyclic blocks (2N-type). <...> Results: If a Hadamard matrix is also a Toeplitz matrix, we show that it must be either cyclic or negacyclic. <...> 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. <...> They are equivalent to Hadamard matrices built from two circulant blocks (2C-type). <...> In an earlier paper [7] Ito proposed a conjecture which is stronger than the famous Hadamard conjecture. <...> It turns out that his conjecture is equivalent to the assertion that the NG-pairs exist for all even lengths. <...> We now describe the content of each of the remaining sections. 2 ИНФОРМАЦИОННОУПРАВЛЯЮЩИЕ СИСТЕМЫ k-Toeplitz matrices: We show that if a Hadamard matrix is also a Toeplitz matrix, then it must be cyclic or negacyclic. <...> As a substitute for Ito’s conjecture we propose the weaker conjecture in which the two negacyclic blocks are replaced by Toeplitz matrices. <...> Three kind of Golay pairs: We define negaperiodic autocorrelation function (NAF) and NG-pairs. <...> The length v must be an even integer or 1. <...> We show that the Turyn multiplication of G-pairs extends to give a multiplication of G-pairs and NGpairs. <...> Cyclic relative difference families: We introduce a natural <...>