  First, we state a generalization of the minimum-distance bound for PIR codes.
Then we describe a construction for linear PIR codes using packing designs and
use it to construct some new 5-PIR codes. Finally, we show that no encoder
(linear or nonlinear) for the binary $r$-th order Hamming code produces a 3-PIR
code except when $r=2$. We use these results to determine the smallest length
of a binary (possibly nonlinear) 3-PIR code of combinatorial dimension up to~6.
A binary 3-PIR code of length 11 and size $2^7$ is necessarily nonlinear, and
we pose the existence of such a code as an open problem.
