For fixed integers r ≥ 2, e ≥ 2, v ≥ r + 1, an r-uniform hypergraph is called G _r (v, e)-free if the union of any e distinct edges contains at least v+1 vertices. Let G _r (n, v, e) denote the maximum number of edges in a G _r (v, e)-free r-uniform hypergraph on n vertices. Brown, Erdós and Sós showed in 1973 that there exist constants c ₁ , c ₂ depending only on r, e, v such that c1n ^er-v/e-1 ≤ f _r (n,v,e) ≤ c2n ^[er-v/e-1] For e - 1|er - v, the lower bound matches the upper bound up to a constant factor; whereas for e - 1 t er - v, it is a notoriously hard problem to determine the correct exponent of n. Our main result is an er-v improvement f _r (n, v, e) = Ω(n e-1 (log n) 1 e-1 ) for any r, e, v satisfying gcd(e - 1, er - v) = 1. Moreover, the hypergraph we constructed is not only gr(v, e)-free but also universally G _r (ir - Γ (i-1)(-1er-v) 1 + i)-free for every 2 <; i <; e. Interestingly, e our new lower bound provides improved constructions for several seemingly unrelated topics in Coding Theory, namely, Parent-Identifying Set Systems, uniform Combinatorial Batch Codes and optimal Locally Recoverable Codes.