Matrix generators for Weil representations

Let $r$ be an odd prime and $\mathbb{F}$ a field containing a primitive $r$th root of unity. Then for all $\mathrm{\ell }\ge 1$, there is a faithful representation $f:{\mathrm{Sp}}_{2\mathrm{\ell }}(r)\to {\mathrm{GL}}_{r^{\mathrm{\ell }}}(\mathbb{F})$ called the Weil representation. We provide explicit matrices generating ${\mathrm{Sp}}_{2\mathrm{\ell }}(r)$ in ${\mathrm{GL}}_{r^{\mathrm{\ell }}}(\mathbb{F})$, which we have implemented in Magma. We also describe such generators for the irreducible Weil representations of ${\mathrm{Sp}}_{2\mathrm{\ell }}(r)$, which are of degree $(r^{\mathrm{\ell }}\pm 1)\mathrm{/}2$ and arise as irreducible constituents of the Weil representations.