  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.

Combinatorics