We present a systematic study of deuterated molecular hydrogen (HD) at high redshift, detected in absorption in the spectra of quasars. We present four new identifications of HD lines associated with known $rm H_2$-bearing Damped Lyman-$alpha$ systems. In addition, we measure upper limits on the $rm HD$ column density in twelve recently identified $rm H_2$-bearing DLAs. We find that the new $rm HD$ detections have similar $N({rm HD})/N(rm H_2)$ ratios as previously found, further strengthening a marked difference with measurements through the Galaxy. This is likely due to differences in physical conditions and metallicity between the local and the high-redshift interstellar media. Using the measured $N({rm HD})/N({rm H_2})$ ratios together with priors on the UV flux ($chi$) and number densities ($n$), obtained from analysis of $rm H_2$ and associated CI lines, we are able to constrain the cosmic-ray ionization rate (CRIR, $zeta$) for the new $rm HD$ detections and for eight known HD-bearing systems where priors on $n$ and $chi$ are available. We find significant dispersion in $zeta$, from a few $times 10^{-18}$ s$^{-1}$ to a few $times 10^{-15}$ s$^{-1}$. We also find that $zeta$ strongly correlates with $chi$ -- showing almost quadratic dependence, slightly correlates with $Z$, and does not correlate with $n$, which probably reflects a physical connection between cosmic rays and star-forming regions.
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