Explicit combinatorial design

Xiongfeng Ma ∗ and Xiaoqing Tan 1, † Center for Quantum Information and Quantum Control, Department of Physics and Department of Electrical & Computer Engineering, University of Toronto, Toronto, Ontario, Canada Department of Mathematics, College of Information Science and Technology, Jinan University, Guangzhou, Guangdong, P. R. China Abstract A combinatorial design is a family of sets that are almost disjoint, which is applied in pseudo random number generations and randomness extractions. The parameter, ρ, quantifying the overlap between the sets within the family, is directly related to the length of a random seed needed and the efficiency of an extractor. Nisan and Wigderson proposed an explicit construction of designs in 1994. Later in 2003, Hartman and Raz proved a bound of ρ ≤ e2 for the Nisan-Wigderson construction. In this work, we prove a tighter bound of ρ < e with a larger parameter range by slightly refining the Nisan-Wigderson construction. Following the block idea used by Raz, Reingold, and Vadhan, we present an explicit weak design with ρ = 1.