Derandomizing Chernoff Bound with Union Bound with an Application to $k$-wise Independent Sets

TL;DR

This paper presents a new explicit derandomization technique for Chernoff bounds using union bounds, leading to efficient constructions of almost k-wise independent sets and establishing near-optimal lower bounds for their size.

Contribution

It introduces an explicit derandomization method that closely matches probabilistic bounds and provides simple polynomial-time constructions of almost k-wise independent sets with tight size bounds.

Findings

Constructed almost k-wise independent sets in polynomial time.

Provided near-tight lower bounds for the size of k-wise independent sets.

Demonstrated the effectiveness of derandomization techniques in combinatorial constructions.

Abstract

Derandomization of Chernoff bound with union bound is already proven in many papers. We here give another explicit version of it that obtains a construction of size that is arbitrary close to the probabilistic nonconstructive size. We apply this to give a new simple polynomial time constructions of almost $k$-wise independent sets. We also give almost tight lower bounds for the size of $k$-wise independent sets.