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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The notion of a Private Information Retrieval (PIR) code was recently introduced by Fazeli, Vardy and Yaakobi [1] who showed that this class of codes permit PIR at reduced levels of storage overhead in comparison with rephcated-server PIR. In the present paper, the construction of an (n, k) τ-server binary linear PIR code having parameters n = lΣ i=0 (m i ), k = (m i ) and τ = 2l for any integer m ≥ l ≥ 0 is presented. These codes are obtained through homogeneous-polynomial evaluation and correspond to the binary. Projective Reed Muller (PRM) code. The construction can be extended to yield PIR codes for any τ = ∊ 2l, 2l − 1 | l ∊ Z, l ≥ 0 and any value of k, through a combination of single-symbol puncturing and shortening of the PRM code. Each of these code constructions above, have smaller storage overhead in comparison with known short block length codes in [1]. For the particular case of τ = 3,4, we show that the codes constructed here are optimal, systematic PIR codes by providing an improved lower bound on the block length n k, τ) of a systematic PIR code. It follows from a result by Vardy and Yaakobi [2], that these codes also yield optimal, systematic primitive multi-set n, k, τ) B batch codes for τ = 3,4. The PIR code constructions presented here also yield upper bounds on the generahzed Hamming weights of binary PRM codes. Consider a large database of @math data items that need to be stored using @math servers. We study how to encode information so that a large number @math of read requests can be performed in parallel while the rate remains constant (and ideally approaches one). This problem is equivalent to the design of multiset Batch Codes introduced by Ishai, Kushilevitz, Ostrovsky and Sahai [17]. We give families of multiset batch codes with asymptotically optimal rates of the form @math and a number of servers @math scaling polynomially in the number of read requests @math . An advantage of our batch code constructions over most previously known multiset batch codes is explicit and deterministic decoding algorithms and asymptotically optimal fault tolerance. Our main technical innovation is a graph-theoretic method of designing multiset batch codes using dense bipartite graphs with no small cycles. We modify prior graph constructions of dense, high-girth graphs to obtain our batch code results. We achieve close to optimal tradeoffs between the parameters for bipartite graph based batch codes. We consider the problem of single-round private information retrieval (PIR) from @math replicated databases. We consider the case when @math databases are outdated (unsynchronized), or even worse, adversarial (Byzantine), and therefore, can return incorrect answers. In the PIR problem with Byzantine databases (BPIR), a user wishes to retrieve a specific message from a set of @math messages with zero-error, irrespective of the actions performed by the Byzantine databases. We consider the @math -privacy constraint in this paper, where any @math databases can collude, and exchange the queries submitted by the user. We derive the information-theoretic capacity of this problem, which is the maximum number of that can be retrieved privately (under the @math -privacy constraint) for every symbol of the downloaded data. We determine the exact BPIR capacity to be @math , if @math . This capacity expression shows that the effect of Byzantine databases on the retrieval rate is equivalent to removing @math databases from the system, with a penalty factor of @math , which signifies that even though the number of databases needed for PIR is effectively @math , the user still needs to access the entire @math databases. The result shows that for the unsynchronized PIR problem, if the user does not have any knowledge about the fraction of the messages that are mis-synchronized, the single-round capacity is the same as the BPIR capacity. Our achievable scheme extends the optimal achievable scheme for the robust PIR (RPIR) problem to correct the introduced by the Byzantine databases as opposed to in the RPIR problem. Our converse proof uses the idea of the cut-set bound in the network coding problem against adversarial nodes. We present a general framework for private information retrieval (PIR) from arbitrary coded databases that allows one to adjust the rate of the scheme to the suspected number of colluding servers. If the storage code is a generalized Reed--Solomon code of length @math and dimension @math , we design PIR schemes that achieve a PIR rate of @math while protecting against any @math colluding servers, for any @math . This interpolates between the previously studied cases of @math and @math and achieves PIR capacity in both of these cases asymptotically as the number of files in the database grows.