Binary Locally Repairable Codes With Large Availability and its Application to Private Information Retrieval

Abstract

Locally Repairable codes (LRCs) have gained significant interest due to applications in distributed storage systems since they enable systems to recover a failed node by accessing few other active nodes. In particular, LRCs with large availability are highly desirable for parallel reading of hot data. In addition to the above applications in distributed storage systems, it was shown by Fazeli et al. that an LRC with large availability can produce a good private information retrieval (PIR) code which allows to reduce the storage overhead of a PIR protocol. Roughly speaking, one can obtain a good PIR code as long as there exists an LRC with large availability. One of the main tasks in studying PIR codes is to design a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> -server PIR code with small length for the given dimension. In particular, the construction of binary PIR codes is of great interest. In this paper, we consider a construction of binary LRCs from polynomial evaluations. As a result, a new class of binary LRCs with large availability are obtained. Applying such LRCs to PIR codes, we obtain a new class of binary PIR codes. On one hand, the binary LRCs constructed are new in the sense that the parameter regime is not covered by the known LRCs. On the other hand, the parameters of PIR codes derived from our LRCs outperform the known results in certain parameter regimes and achieve the lower bound given by Fazeli et al. up to an absolute constant.