Inflation of Perfect Arrays Over the Basic Quaternions of Size $mn=(q+1)/2$

Abstract

Arasu and de Launey showed that every perfect quaternary array (that is perfect arrays over the four roots of unity $\pm 1,\pm i$) of size $m\times n$, can be inflated into another perfect quaternary array of size $mp\times np$, provided $p=mn-1$ is s prime number. Likewise, they showed that every perfect quaternary array of size $m\times n$, can be inflated into another perfect quaternary array of size $mq\times nq$, provided $q=2mn-1$ is a prime number and $q\equiv 3(mod\ 4)$. Following from Arasu and de Launey’s first construction, Barrera Acevedo and Jolly showed that every perfect array over the basic quaternions, $\{1,-1,i,-i,j,-j,k,-k\}$, of sizes $m\times n$, can be inflated into a new perfect array over the basic quaternions of size $mp\times np$, provided $p=mn-1$ is s prime number. Combining this construction with the existence of infinitely many modified Lee sequences over $\{1,-1,i,-i,j\}$ (in the sense of Barrera Acevedo and Hall), they showed the existence of infinitely many perfect arrays over the basic quaternions, with appearances of all the basic quaternion elements $1,-1,i,-i,j,-j,k$ and $-k$. In this work, we show that every perfect array over the basic quaternions, of size $m\times n$, can be inflated into a perfect quaternary array of size $mq\times nq$, provided $q=2mn-1$ is a prime number and $q\equiv 3(mod\ 4)$.