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A construction for bipatite Turán numbers

We consider in detail the well-known family of graphs $G(q,t)$ that establish an asymptotic lower bound for Turán numbers $\mathrm{ex}(n,K_{2,t+1})$. We prove that $G(q,t)$ for some specific $q$ and $t$ also gives an asymptotic bound for $K_{3,3}$ and for some higher complete bipartite graphs as well. The asymptotic bounds we prove are the same as provided by the well-known Norm-graphs.