  High-stress Si$_3$N$_4$ nanoresonators have become an attractive choice for
electro- and optomechanical devices. Membrane resonators can achieve quality
factor ($Q$) - frequency ($f$) products exceeding $10^{13}$ Hz, enabling (in
principle) quantum coherent operation at room temperature. String-like beam
resonators possess conventionally 10 times smaller $Q\cdot f$ products;
however, on account of their much larger $Q$-to-mass ratio and reduced mode
density, they remain a canonical choice for precision force, mass, and charge
sensing, and have recently enabled Heisenberg-limited position measurements at
cryogenic temperatures. Here we explore two techniques to enhance the
$Q$-factor of a nanomechanical beam. The techniques relate to two main loss
mechanisms: internal loss, which dominates for large aspect ratios and
$f\lesssim100$ MHz, and radiation loss, which dominates for small aspect ratios
and $f\gtrsim100$ MHz. First we show that by embedding a nanobeam in a 1D
phononic crystal, it is possible to localize its flexural motion and shield it
against radiation loss. Using this method, we realize $f>100$ MHz modes with
$Q\sim 10^4$, consistent with internal loss and contrasting sharply with
unshielded beams of similar dimensions. We then study the $Q\cdot f$ products
of high-order modes of mm-long nanobeams. Taking advantage of the mode-shape
dependence of stress-induced `loss-dilution', we realize a $f\approx 4$ MHz
mode with $Q\cdot f\approx9\cdot 10^{12}$ Hz. Our results can extend room
temperature quantum coherent operation to ultra-low-mass 1D nanomechanical
oscillators.
