Extractors for adversarial sources via extremal hypergraphs

A note on explicit constructions of designs

“…Proof of Theorem 1.6. We prove this theorem by induction on i∈ [4] ℓ i . Theorem 1.5 shows that the base case i∈ [4] ℓ i = 0 holds, so we may assume that i∈ [4] ℓ i ≥ 1.…”

“…We prove this theorem by induction on i∈ [4] ℓ i . Theorem 1.5 shows that the base case i∈ [4] ℓ i = 0 holds, so we may assume that i∈ [4] ℓ i ≥ 1. Let s = 3 ℓ 1 4 ℓ 2 5 ℓ 3 6 ℓ 4 + 1, and let us assume, for the sack of simplicity, that ℓ 1 ≥ 1 (the other cases can be proved using a similar argument).…”

“…For example, in the seminal work of Nisan and Wigderson [10], dense (n, r, s)-systems are used to construct pseudorandom generators (PRGs) (see also [17,12] for more applications). More recently, explicit constructions of (n, r, s)-systems with small independence number were used to construct extractors for adversarial sources [4,3].…”

“…Rödl and Šiňajová's proof of the existence of an (n, r, s)-system with small independence number uses the Lovász local lemma, and hence it does not provide an explicit way to construct them. Perhaps the first explicit construction of an (n, 3, 2)-system (also called a Steiner triple system) with independence number O(n 1−ǫ ) for some absolute constant ǫ > 0 is due to Chattopadhyay, Goodman, Goyal, and Li [4]. Their proof uses results about cap sets (see [5,6]).…”

“…Theorem 1.3 (Chattopadhyay-Goodman-Goyal-Li [4]). There exists a constant C ≥ 1 such that for every integer n ≥ 3 there exists an explicit construction of an (n, 3, 2)-system with independence number at most Cn 0.9228 .…”