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.
A special case of @math batch and PIR codes, where the size of the recovery sets @math is not restricted (for example, it can be assumed that @math ), is studied in @cite_1 . Let @math be the shortest length @math of any systematic linear batch code with unrestricted size of the recovery set, and @math be the shortest length @math of any linear systematic PIR code with unrestricted size of the recovery set. Then the optimal redundancy of batch and PIR codes, respectively, is defined as @math and @math .
@cite_1
2591083178 2951944286 2621688363 2168831133
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. The goal of (stable) sparse recovery is to recover a @math -sparse approximation @math of a vector @math from linear measurements of @math . Specifically, the goal is to recover @math such that @math for some constant @math and norm parameters @math and @math . It is known that, for @math or @math , this task can be accomplished using @math non-adaptive measurements CRT06:Stable-Signal and that this bound is tight DIPW, FPRU, PW11 . In this paper we show that if one is allowed to perform measurements that are adaptive , then the number of measurements can be considerably reduced. Specifically, for @math and @math we show A scheme with @math measurements that uses @math rounds. This is a significant improvement over the best possible non-adaptive bound. A scheme with @math measurements that uses two rounds. This improves over the best possible non-adaptive bound. To the best of our knowledge, these are the first results of this type.