  The universal asymptotic amplitude ratio between the gyration radius and the
hydrodynamic radius of self-avoiding walks is estimated by high-resolution
Monte Carlo simulations. By studying chains of length of up to $N = 2^{25}
\approx 34 \times 10^6$ monomers, we find that the ratio takes the value
$R_{\mathrm{G}}/R_{\mathrm{H}} = 1.5803940(45)$, which is several orders of
magnitude more accurate than the previous state of the art. This is facilitated
by a sampling scheme which is quite general, and which allows for the efficient
estimation of averages of a large class of observables. The competing
corrections to scaling for the hydrodynamic radius are clearly discernible. We
also find improved estimates for other universal properties that measure the
chain dimension. In particular, a method of analysis which eliminates the
leading correction to scaling results in a highly accurate estimate for the
Flory exponent of $\nu = 0.58759700(40)$.
