Projects per year
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} ∪ qQ_{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} ∪ qQ_{8} yields a Hadamard matrix of order 4n (and a quaternionic Hadamard matrix of order n over Q_{8} ∪ qQ_{8}).
Original language  English 

Pages (fromto)  357368 
Number of pages  12 
Journal  Cryptography and Communications: Discrete Structures, Boolean Functions and Sequences 
Volume  10 
Issue number  2 
DOIs  
Publication status  Published  1 Mar 2018 
Keywords
 Hadamard matrices
 Perfect sequences
 Quaternions
 Relative difference sets
Projects
 2 Finished

Computing with matrix groups and Lie algebras: new concepts and applications
Australian Research Council (ARC)
1/02/14 → 1/02/17
Project: Research

Planar Brownian motion and complex analysis
Australian Research Council (ARC)
2/01/14 → 11/01/17
Project: Research