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arXiv · 1508.00640

Negaperiodic Golay pairs and Hadamard matrices

N. A. Balonin·D. Z. Djokovic
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Abstract

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.

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BibTeXRIS

N. A. Balonin, D. Z. Djokovic. 2015-08-04. Negaperiodic Golay pairs and Hadamard matrices. https://doi.org/10.15217/issn1684-8853.2015.5.2

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