Explicit Extremal Designs and Applications to Extractors

An $(n,r,s)$-design, or $(n,r,s)$-partial Steiner system, is an $r$-uniform hypergraph over $n$ vertices with pairwise hyperedge intersections of size $0$, we extract from $(N,K,n,k)$-adversarial sources of locality $0$, where $K\geq N^\delta$ and $k\geq\text{polylog }n$. The previous best result (Chattopadhyay et al., STOC 2020) required $K\geq N^{1/2+o(1)}$. As a result, we get extractors for small-space sources over $n$ bits with entropy requirement $k\geq n^{1/2+\delta}$, whereas the previous best result (Chattopadhyay et al., STOC 2020) required $k\geq n^{2/3+\delta}$.