Given a database, the private information retrieval (PIR) protocol allows a user to make queries to several servers and retrieve a certain item of the database via the feedbacks, without revealing the privacy of the specific item to any single server. Classical models of PIR protocols require that each server stores a whole copy of the database. Recently new PIR models are proposed with coding techniques arising from distributed storage system. In these new models each server only stores a fraction $1/s$ of the whole database, where $s>1$ is a given rational number. PIR array codes are recently proposed by Fazeli, Vardy and Yaakobi to characterize the new models. Consider a PIR array code with $m$ servers and the $k$-PIR property (which indicates that these $m$ servers may emulate any efficient $k$-PIR protocol). The central problem is to design PIR array codes with optimal rate $k/m$. Our contribution to this problem is three-fold. First, for the case $1
2$, we derive a new upper bound on the rate of a PIR array code. Finally, for the case $s>2$, we analyze a new construction by Blackburn and Etzion and show that its rate is better than all the other existing constructions.