Abstarct: For fixed integers r ≥ 2, e ≥ 2, v ≥ r + 1 an r-uniform hypergraph is calledg_r (v,e) -free if the union of any e distinct edges contains at least v + 1 vertices. Letf_r (n,v,e) denote the maximum number of edges in ag_r (v,e)-free r-uniform hypergraph on n vertices. Brown, Erdős and Sós showed in 1973 that there exist constants c_1, c_2depending only on r,e,v such that c_1 n^((ev-v)/(e-1))≤f_r (n,v,e)≤c_2 n^⌈(ev-V)/(e-1)⌉ For e-1 | er-v the lower bound matches the upper bound up to a constant factor; whereas for e - 1 ∤ er - v, it is a notoriously hard problem to determine the correct exponent of n for general r,v,e. Our main result is an improvement on the above lower bound by a(log⁡n )^(1/(e-1)) factor f_r (n,v,e)=Ω(n^((er-v)/(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 g_r (v,e)-free but also universallyg_r (ir-⌈(i-1)(er-v)/(e-1)⌉,i) -free for every 2 ≤ i ≤ e. Interestingly, our new lower bound provides improved constructions for several seemingly unrelated objects in Coding Theory, namely, Parent-Identifying Set Systems, uniform Combinatorial Batch Codes and optimal Locally Recoverable Codes. The proof of the main result is baxsed on a novel application of the well-known lower bound on the hypergraph independence number due to Ajtai, Komlós, Pintz, Spencer, and Szemerédi, and Duke, Lefmann, and Rödl.