Perfect sequences over the quaternions and (4n, 2, 4n, 2n)-relative difference sets in C n × Q 8

Abstract

Perfect sequences over general quaternions were introduced in 2009 by Kuznetsov. The existence of perfect sequences of increasing lengths over the basic quaternions Q 8 = {±1, ±i, ±j, ±k} was established in 2012 by Barrera Acevedo and Hall. The aim of this paper is to prove a 1–1 correspondence between perfect sequences of length n over Q 8 ∪ q Q 8 with q = (1 + i + j + k)/2, and (4n, 2, 4n, 2n)-relative difference sets in C n × Q 8 with forbidden subgroup C 2; here C m is a cyclic group of order m. We show that if n = p a + 1 for a prime p and integer a ≥ 0 with n ≡ 2 mod 4, then there exists a (4n, 2, 4n, 2n)-relative different set in C n × Q 8 with forbidden subgroup C 2. Lastly, we show that every perfect sequence of length n over Q 8 ∪ q Q 8 yields a Hadamard matrix of order 4n (and a quaternionic Hadamard matrix of order n over Q 8 ∪ q Q 8).