  It was recently shown by Fazeli et al. that the storage overhead of a
traditional $t$-server private information retrieval (PIR) protocol can be
significantly reduced using the concept of a $t$-server PIR code. In this work,
we show that a family of $t$-server PIR codes (with increasing dimensions and
blocklengths) can be constructed from an existing $t$-server PIR code through
lengthening by a single information symbol and code extension by at most
$\bigl\lceil t/2\bigr\rceil$ code symbols. Furthermore, by extending a code
construction notion from Steiner systems by Fazeli et al., we obtain a specific
family of $t$-server PIR codes. Based on a code construction technique that
lengthens and extends a $t$-server PIR code simultaneously, a basic algorithm
to find good (i.e., small blocklength) $t$-server PIR codes is proposed. For
the special case of $t=5$, we find provably optimal PIR codes for code
dimensions $k\leq 6$, while for all $7\leq k\leq 32$ we find codes of smaller
blocklength than the best known codes from the literature. Furthermore, in the
case of $t = 8$, we also find better codes for $k = 5, 6, 11, 12$. Numerical
results show that most of the best found $5$-server PIR codes can be
constructed from the proposed family of codes connected to Steiner systems.
