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.
In this work, we study (primitive, multiset) batch codes with restricted size of the recovery sets, as they are defined in @cite_7 (see also @cite_1 ).
@cite_1 @cite_7
2951572681 2951944286 2967056459 1996042140
In this paper, we study codes with locality that can recover from two erasures via a sequence of two local, parity-check computations. By a local parity-check computation, we mean recovery via a single parity-check equation associated to small Hamming weight. Earlier approaches considered recovery in parallel; the sequential approach allows us to potentially construct codes with improved minimum distance. These codes, which we refer to as locally 2-reconstructible codes, are a natural generalization along one direction, of codes with all-symbol locality introduced by Gopalan , in which recovery from a single erasure is considered. By studying the Generalized Hamming Weights of the dual code, we derive upper bounds on the minimum distance of locally 2-reconstructible codes and provide constructions for a family of codes based on Tur 'an graphs, that are optimal with respect to this bound. The minimum distance bound derived here is universal in the sense that no code which permits all-symbol local recovery from @math erasures can have larger minimum distance regardless of approach adopted. Our approach also leads to a new bound on the minimum distance of codes with all-symbol locality for the single-erasure case. 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. In this paper, we study codes with locality that can recover from two erasures via a sequence of two local, parity-check computations. By a local parity-check computation, we mean recovery via a single parity-check equation associated with small Hamming weight. Earlier approaches considered recovery in parallel; the sequential approach allows us to potentially construct codes with improved minimum distance. These codes, which we refer to as locally 2-reconstructible codes, are a natural generalization along one direction, of codes with all-symbol locality introduced by , in which recovery from a single erasure is considered. By studying the generalized Hamming weights of the dual code, we derive upper bounds on the minimum distance of locally 2-reconstructible codes and provide constructions for a family of codes based on Turan graphs, that are optimal with respect to this bound. The minimum distance bound derived here is universal in the sense that no code which permits all-symbol local recovery from 2 erasures can have larger minimum distance regardless of approach adopted. Our approach also leads to a new bound on the minimum distance of codes with all-symbol locality for the single-erasure case. We design flexible schemes to explore the tradeoffs between storage space and access efficiency in reliable data storage systems. Aiming at this goal, two new classes of erasure-resilient codes are introduced -- Basic Pyramid Codes (BPC) and Generalized Pyramid Codes (GPC). Both schemes require slightly more storage space than conventional schemes, but significantly improve the critical performance of read during failures and unavailability. As a by-product, we establish a necessary matching condition to characterize the limit of failure recovery, that is, unless the matching condition is satisfied, a failure case is impossible to recover. In addition, we define a maximally recoverable (MR) property. For all ERC schemes holding the MR property, the matching condition becomes sufficient, that is, all failure cases satisfying the matching condition are indeed recoverable. We show that GPC is the first class of non-MDS schemes holding the MR property.