Negaperiodic Golay pairs and Hadamard matrices

TL;DR

This paper introduces negaperiodic Golay pairs, explores their connection to Hadamard matrices, and extends Golay pair multiplication, linking the existence of negaperiodic Golay pairs to Ito's conjecture.

Contribution

It defines negaperiodic Golay pairs, establishes their relation to Hadamard matrices, and generalizes Golay pair multiplication, connecting these concepts to Ito's conjecture.

Findings

Negaperiodic Golay pairs are equivalent to certain Hadamard matrices.

Turyn multiplication extends to negaperiodic Golay pairs.

Existence of negaperiodic Golay pairs for all even lengths relates to Ito's conjecture.

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.