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

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