Lengthening and Extending Binary Private Information Retrieval Codes

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 $\lceil t\mathrm{/}2\rceil$ 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 $k\le 6$, while for all $7\le k\le 32$ 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.