Let $C$ be the set of all vectors of dimension $n$ such that each of its entries are one of $-1$, $0$ and $1$ and also that the every $v \in C$ has at least $\frac{n}{100}$ $1$'s and at least $\frac{n}{100}$ $-1$'s. For a matrix $A$ (its dimensions are $\Lambda n \times n$ for some sufficiently large constant $\Lambda$), define $||A||_{C,\infty}$ to be

$||A||_{C,\infty} = min _{x \in C} \ \ \ ||A x|| _{\infty}$

Is it possible to construct an $A$ such that all its entries come from the interval $[-1,1]$ and $||A||_{C,\infty} = n^{1/2 + \epsilon}$ for some $\epsilon>0$. Constructing the matrix $A$ randomly by choosing each of its entries to be independent and uniformly random elements in $[-1,1]$ gets $||A|| _{C,\infty} =n^{1/2}$. The construction does not need to be explicit.