  A $k$-server Private Information Retrieval (PIR) code is a binary linear
$[m,s]$-code admitting a generator matrix such that for every integer $i$ with
$1\le i\le s$ there exist $k$ disjoint subsets of columns (called recovery
sets) that add up to the vector of weight one, with the single $1$ in position
$i$. As shown in \cite{Fazeli1}, a $k$-server PIR code is useful to reduce the
storage overhead of a traditional $k$-server PIR protocol. Finding $k$-server
PIR codes with a small blocklength for a given dimension has recently become an
important research challenge. In this work, we propose new constructions of PIR
codes from combinatorial structures, introducing the notion of $k$-partial
packing. Several bounds over the existing literature are improved.
