Self-dual codes over ${\mathbb{F}}_5$ and $s$-extremal unimodular lattices

New $s$-extremal extremal unimodular lattices in dimensions $38$, $40$, $42$ and $44$ are constructed from self-dual codes over ${\mathbb{F}}_5$ by Construction A. In the process of constructing these codes, we obtain a self-dual $[44,22,14]$ code over ${\mathbb{F}}_5$. In addition, the code implies a $[43,22,13]$ code over ${\mathbb{F}}_5$. These codes have larger minimum weights than the previously known $[44,22]$ codes and $[43,22]$ codes, respectively.