Derandomized construction of combinatorial batch codes

Abstract

Combinatorial Batch Codes (CBCs), replication-based variant of Batch Codes introduced by Ishai et al. in [IKOS04], abstracts the following data distribution problem: n data items are to be replicated among m servers in such a way that any k of the n data items can be retrieved by reading at most one item from each server with the total amount of storage over m servers restricted to N. Given parameters m, c, and k, where c and k are constants, one of the challenging problems is to construct c-uniform CBCs (CBCs where each data item is replicated among exactly c servers) which maximizes the value of n. In this work, we present explicit construction of c-uniform CBCs with Ω(mc−1+1/k) data items. The construction has the property that the servers are almost regular, i. e., number of data items stored in each server is in the range [nc/m −√n/2 ln(4m), nc/m +√n/2 ln(4m)]. The construction is obtained through better analysis and derandomization of the randomized construction presented in [IKOS04]. Analysis reveals almost regularity of the servers, an aspect that so far has not been addressed in the literature. The derandomization leads to explicit construction for a wide range of values of c (for given m and k) where no other explicit construction with similar parameters, i. e., with n = Ω(mc−1+1/k), is known. Finally, we discuss possibility of parallel derandomization of the construction.