Mumford representation and Riemann Roch space of a divisor on a hyperelliptic curve

For an (imaginary) hyperelliptic curve $\mathcal{H}$ of genus $g$, with a Weierstrass point $\mathrm{\Omega }$, taken as the point at infinity, we determine a basis of the Riemann-Roch space $\mathcal{L}(\mathrm{\Delta }+m\mathrm{\Omega })$, where $\mathrm{\Delta }$ is of degree zero, directly from the Mumford representation of $\mathrm{\Delta }$. This provides in turn a generating matrix of a Goppa code.