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 tserver 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 ⌈ t/2 ⌉ 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 tserver 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 ≤ 6, while for all 5 ≤ k ≤ 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.