Explicit bases of the Riemann-Roch spaces on divisors on hyperelliptic curves

TL;DR

This paper explicitly constructs bases for Riemann-Roch spaces on hyperelliptic curves and applies the results to generate Goppa codes, enhancing algebraic coding theory methods.

Contribution

It provides explicit bases for Riemann-Roch spaces on hyperelliptic curves with specific divisors, and demonstrates their application in coding theory.

Findings

Explicit bases for Riemann-Roch spaces on hyperelliptic curves

Construction of generator matrices for Goppa codes

Application to codes with genus 3 and degree 4

Abstract

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 $P_1+\cdots+ P_j+(n-j)\Omega$, with $0 \le j \le g$, where $\Omega$ 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 $n=4.$