Optimal Possibly Nonlinear 3-PIR Codes of Small Size

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 27 is necessarily nonlinear (as a PIR code), and we pose the existence of such a code as an open problem.