Explicit bases of the Riemann-Roch spaces on divisors on hyperelliptic curves
This work addresses a foundational problem in algebraic geometry and coding theory, offering incremental advancements in constructing explicit bases for divisors on hyperelliptic curves.
The paper tackles the problem of determining explicit bases for Riemann-Roch spaces on divisors of hyperelliptic curves, and as a result, it provides a basis for divisors with positive degree and applies this to construct a generator matrix for a specific Goppa code with parameters j=g=3 and n=4.
For an (imaginary) hyperelliptic curve $\mathcal{H}$ of genus $g$, we determine a basis of the Riemann-Roch space $\mathcal{L}(D)$, where $D$ is a divisor with positive degree $n$, linearly equivalent to $P_1+\cdots+ P_j+(n-j)Ω$, with $0 \le j \le g$, where $Ω$ is a Weierstrass point, taken as the point at infinity. As an application, we determine a generator matrix of a Goppa code for $j=g=3$ and $n=4.$