Almost optimum $\ell$-covering of $\mathbb{Z}_n$

A subset $B$ of ring $\mathbb{Z}_n$ is called a $\ell$-covering set if $\{ ab \pmod n \mid 0\leq a \leq \ell, b\in B\} = \mathbb{Z}_n$. We show there exists a $\ell$-covering set of $\mathbb{Z}_n$ of size $O(\frac{n}{\ell}\log n)$ for all $n$ and $\ell$, and how to construct such set. We also show examples where any $\ell$-covering set must have size $\Omega(\frac{n}{\ell}\frac{\log n}{\log \log n})$. The proof uses a refined bound for relative totient function obtained through sieve theory, and existence of a large divisor with linear divisor sum.