  We have conducted the most sensitive low frequency (below 100 MHz) search to
date for prompt, low-frequency radio emission associated with short-duration
gamma-ray bursts (GRBs), using the Owens Valley Radio Observatory Long
Wavelength Array (OVRO-LWA). The OVRO-LWA's nearly full-hemisphere
field-of-view ($\sim20$,$000$ square degrees) allows us to search for
low-frequency (sub-$100$ MHz) counterparts for a large sample of the subset of
GRB events for which prompt radio emission has been predicted. Following the
detection of short GRB 170112A by Swift, we used all-sky OVRO-LWA images
spanning one hour prior to and two hours following the GRB event to search for
a transient source coincident with the position of GRB 170112A. We detect no
transient source, with our most constraining $1\sigma$ flux density limit of
$650~\text{mJy}$ for frequencies spanning $27~\text{MHz}-84~\text{MHz}$. We
place constraints on a number of models predicting prompt, low-frequency radio
emission accompanying short GRBs and their potential binary neutron star merger
progenitors, and place an upper limit of $L_\text{radio}/L_\gamma \lesssim
7\times10^{-16}$ on the fraction of energy released in the prompt radio
emission. These observations serve as a pilot effort for a program targeting a
wider sample of both short and long GRBs with the OVRO-LWA, including bursts
with confirmed redshift measurements which are critical to placing the most
constraining limits on prompt radio emission models, as well as a program for
the follow-up of gravitational wave compact binary coalescence events detected
by advanced LIGO and Virgo.
