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NEGAPERIODIC GOLAY PAIRS AND HADAMARD MATRICES

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). The Hadamard matrices of 2N-type are equivalent to negaperiodic Golay pairs. 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. Practical relevance: Hadamard matrices have direct practical applications to the problems of noise-immune coding and compression and masking of video information.

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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). <...> 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. <...> Introduction The Golay pairs (abbreviated as G-pairs, and also known as Golay sequences) have been introduced in a note of M. Golay [1] published in 1961. <...> It is now known that periodic Golay pairs exist for infinitely many lengths for which no ordinary Golay pairs are known [5]. <...> In this paper we complete the picture by defining the negaperiodic Golay pairs (NG-pairs). <...> The NG-pairs were first introduced by N. Ito, under the name of “associated pairs”, in his paper [6] published in 2000. <...> An intereseting observation is that the ordinary Golay pairs are precisely the pairs which are both PG and NG-pairs. <...> 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. <...> This is drastically different from the known facts about ordinary and periodic Golay pairs. <...> As far as we know, no NG-pairs of length 94 have been constructed <...>
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