Negaperiodic Golay pairs and Hadamard matrices

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.