Perfect sequences over the quaternions and (4n,2,4n,2n)-relative difference sets in C_n × Q_8

摘要

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) boolean OR qQ(8) with q = (1 + i + j + k)/2, and (4n, 2, 4n, 2n)-relative difference sets in C-n x 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 x Q(8) with forbidden subgroup C-2. Lastly, we show that every perfect sequence of length n over Q(8) boolean OR q Q(8) yields a Hadamard matrix of order 4n (and a quaternionic Hadamard matrix of order n over Q(8) boolean OR qQ(8)).