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

Abstract

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