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We present the discovery of the optical afterglow and host galaxy of the {\itSwift} short-duration gamma-ray burst, GRB\,181123B. Observations withGemini-North starting at $\approx 9.1$~hr after the burst reveal a faintoptical afterglow with $i\approx25.1$~mag, at an angular offset of 0.59 $\pm$0.16$''$ from its host galaxy. Using $grizYJHK$ observations, we measure aphotometric redshift of the host galaxy of $z = 1.77^{+0.30}_{-0.17}$. From acombination of Gemini and Keck spectroscopy of the host galaxy spanning4500-18000~\AA , we detect a single emission line at 13390~\AA, inferred asH$\beta$ at $z = 1.754 \pm 0.001$ and corroborating the photometric redshift.The host galaxy properties of GRB\,181123B are typical to those of other SGRBhosts, with an inferred stellar mass of $\approx 1.7 \times10^{10}\,M_{\odot}$, mass-weighted age of $\approx 0.9$~Gyr and opticalluminosity of $\approx 0.9L^{*}$. At $z=1.754$, GRB\,181123B is the mostdistant secure SGRB with an optical afterglow detection, and one of only threeat $z>1.5$. Motivated by a growing number of high-$z$ SGRBs, we explore theeffects of a missing $z>1.5$ SGRB population among the current {\it Swift}sample on delay time distribution models. We find that log-normal models withmean delay times of $\approx 4-6$~Gyr are consistent with the observeddistribution, but can be ruled out to $95\%$ confidence with an additional$\approx1-5$~{\it Swift} SGRBs recovered at $z>1.5$. In contrast, power-lawmodels with $\propto$ $t^{-1}$ are consistent with the redshift distributionand can accommodate up to $\approx30$ SGRBs at these redshifts. Under thismodel, we predict that $\approx 1/3$ of the current {\it Swift} population ofSGRBs is at $z>1$. The future discovery or recovery of existing high-$z$ SGRBswill provide significant discriminating power on their delay timedistributions, and thus their formation channels.

Discovery of the optical afterglow and host galaxy of short GRB181123B at $z =1.754$: Implications for Delay Time Distributions