Linear batch codes and codes for private information retrieval (PIR) with a query size @math and a restricted size @math of the reconstruction sets are studied. New bounds on the parameters of such codes are derived for small values of @math or of @math by providing corresponding constructions. By building on the ideas of Cadambe and Mazumdar, a new bound in a recursive form is derived for batch codes and PIR codes.
There is a number of bounds on the parameters of batch codes in the literature, but it is often difficult to make a comparison due to slight variations in the models and assumptions made. Thus, it is proven in @cite_7 that for a linear @math batch code over @math , In particular, when the code is systematic, the bound can be tighten a bit, as follows: If the queries in Definition are restricted to @math , then the corresponding code is called an @math code for private information retrieval (PIR) @cite_12 , or simply @math PIR code. In particular, all batch codes are PIR codes with the corresponding parameters. It should be mentioned that the proofs of ) and ) in @cite_7 work in analogous way for the PIR codes too, and therefore these two bounds hold for general and systematic @math PIR codes, respectively.
@cite_12 @cite_7
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Private information retrieval (PIR) protocols allow a user to retrieve a data item from a database without revealing any information about the identity of the item being retrieved. Specifically, in information-theoretic @math -server PIR, the database is replicated among @math non-communicating servers, and each server learns nothing about the item retrieved by the user. The cost of PIR protocols is usually measured in terms of their communication complexity, which is the total number of bits exchanged between the user and the servers, and storage overhead, which is the ratio between the total number of bits stored on all the servers and the number of bits in the database. Since single-server information-theoretic PIR is impossible, the storage overhead of all existing PIR protocols is at least @math . In this work, we show that information-theoretic PIR can be achieved with storage overhead arbitrarily close to the optimal value of @math , without sacrificing the communication complexity. Specifically, we prove that all known @math -server PIR protocols can be efficiently emulated, while preserving both privacy and communication complexity but significantly reducing the storage overhead. To this end, we distribute the @math bits of the database among @math servers, each storing @math coded bits (rather than replicas). For every fixed @math , the resulting storage overhead @math approaches @math as @math grows; explicitly we have @math . Moreover, in the special case @math , the storage overhead is only @math . In order to achieve these results, we introduce and study a new kind of binary linear codes, called here @math -server PIR codes. We then show how such codes can be constructed, and we establish several bounds on the parameters of @math -server PIR codes. Finally, we briefly discuss extensions of our results to nonbinary alphabets, to robust PIR, and to @math -private PIR. We present new upper bounds on the parameters of batch codes with restricted query size. These bounds are an improvement on the Singleton bound. The techniques for derivations of these bounds are based on the ideas in the literature for codes with locality. By employing additional ideas, we obtain further improvements, which are specific for batch codes.