Constructions of batch codes with near-optimal redundancy

Batch codes, first studied by Ishai et al., are a coding scheme to encode n information bits into m buckets, in a way that every batch request of k bits can be decoded while at most one bit is read from each bucket. In this work we study the class of multiset primitive batch codes, in which every bucket stores a single bit and bits can be requested multiple times. We simply refer to these codes as batch codes. The main problem under this paradigm is to optimize the number of encoded bits, which is the number of buckets, for given n and k, and we denote this value by B(n, k). Since there are several asymptotically optimal constructions of these codes, we are motivated to evaluate their optimality by their redundancy. Thus we define the optimal redundancy of batch codes to be r_B(n, k) t B(n, k) - n. Our main result in this paper claims that for any fixed k, r_B(n, k) = O(√n log(n)).