Perfect arrays of unbounded sizes over the basic quaternions

Abstract

We show the existence of perfect arrays, of unbounded sizes, over the basic quaternions {1,???1,i,???i,j,???j,k,???k}. We translate the algorithm of Arasu and de Launey, to inflate perfect arrays over the four roots of unity, from a polynomial, into a simple matrix approach. Then, we modify this algorithm to inflate perfect arrays over the basic quaternions {1,???1,i,???i,j,???j,k,???k}. We show that all modified Lee Sequences (in the sense of Barrera Acevedo and Hall, Lect Notes Comput Sci 159---167, 2012) of length m?=?p?+?1???2 (mod 4), where p is a prime number, can be folded into a perfect two-dimensional array (with only one occurrence of the element j) of size $2\times \frac{m}{2}$ , with $GCD(2,\frac{m}{2})=1$ . Then, each of these arrays can be inflated into perfect arrays of sizes $2p\times \frac{m}{2}p$ (previously unknown sizes), with a random appearance of all the elements 1,???1,i,???i,j,???j,k,???k.