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
2951944286 1893105184
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 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.