Sparse Hypergraphs with Applications to Coding Theory

For fixed integers $r\ge 3,e\ge 3,v\ge r+1$, an $r$-uniform hypergraph is called $\mathscr{G}_r(v,e)$-free if the union of any $e$ distinct edges contains at least $v+1$ vertices. Brown, Erd\H{o}s and S\'{o}s showed that the maximum number of edges of such a hypergraph on $n$ vertices, denoted as $f_r(n,v,e)$, satisfies $$\Omega(n^{\frac{er-v}{e-1}})=f_r(n,v,e)=\mathcal{O}(n^{\lceil\frac{er-v}{e-1}\rceil}).$$ For $e-1\mid er-v$, the lower bound matches the upper bound up to a constant factor; whereas for $e-1\nmid er-v$, in general it is a notoriously hard problem to determine the correct exponent of $n$. Among other results, we improve the above lower bound by showing that $$f_r(n,v,e)=\Omega(n^{\frac{er-v}{e-1}}(\log n)^{\frac{1}{e-1}})$$ for any $r,e,v$ satisfying $\gcd(e-1,er-v)=1$. The hypergraph we constructed is in fact $\mathscr{G}_r(ir-\lceil\frac{(i-1)(er-v)}{e-1}\rceil,i)$-free for every $2\le i\le e$, and it has several interesting applications in Coding Theory. The proof of the new lower bound is based on a novel application of the lower bound on the hypergraph independence number due to Duke, Lefmann, and R{\"o}dl.