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.
The @math th coordinate of an @math code is said to have locality @math and availability @math if there exist @math disjoint groups, each containing at most @math other coordinates that can together recover the value of the @math th coordinate. This property is particularly useful for codes for distributed storage systems because it permits local repair and parallel accesses of hot data. In this paper, for any positive integers @math and @math , we construct a binary linear code of length @math which has locality @math and availability @math for all coordinates. The information rate of this code attains @math , which is always higher than that of the direct product code, the only known construction that can achieve arbitrary locality and availability.