Perfect arrays of unbounded sizes over the basic quaternions

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 159167, 2012) of length m = p + 1 equivalent to 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 x m/2, with GCD(2, m/2) = 1. Then, each of these arrays can be inflated into perfect arrays of sizes 2p x m/2 p (previously unknown sizes), with a random appearance of all the elements 1,-1, i,-i, j,-j, k,-k.