How many miles from $L_{\mathrm{\infty }}$ to ${\mathrm{\ell }}_{\mathrm{\infty }}$?

The classical Banach spaces $L_{\mathrm{\infty }}[0,1]$ and ${\mathrm{\ell }}_{\mathrm{\infty }}$ are isomorphic. We present here some lower and upper bounds for their Banach-Mazur distance.