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

Giovanni Falcone, Giuseppe Filippone

TL;DR

The paper addresses the problem of deriving an explicit, basis-friendly description of the Riemann–Roch space $\mathcal{L}(D)$ for $D=\Delta+m\Omega$ on a hyperelliptic curve directly from the Mumford representation $\Delta=\operatorname{div}(u,v)$ of a degree-zero divisor. It proves a main theorem giving concrete bases in two regimes depending on $m$ and $t=\deg u$, using the function $\Psi(x,y)$ derived from $u$ and $v$ (and $h$ in characteristic $2$), and provides precise dimension formulas. This enables direct construction of generating matrices for Goppa codes on hyperelliptic curves, without requiring explicit knowledge of the curve containing the divisor's support. The approach is illustrated with an explicit toy example over $\mathrm{GF}(101)$ that yields a $[10,5,6]$ MDS code and demonstrates how to realize AG-Goppa codes by selecting appropriate $u,v$ and a corresponding curve equation. Overall, the work offers a practical, data-efficient method to obtain Riemann–Roch bases and to deploy hyperelliptic-curve codes in coding applications.

Abstract

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

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

TL;DR

The paper addresses the problem of deriving an explicit, basis-friendly description of the Riemann–Roch space for on a hyperelliptic curve directly from the Mumford representation of a degree-zero divisor. It proves a main theorem giving concrete bases in two regimes depending on and , using the function derived from and (and in characteristic ), and provides precise dimension formulas. This enables direct construction of generating matrices for Goppa codes on hyperelliptic curves, without requiring explicit knowledge of the curve containing the divisor's support. The approach is illustrated with an explicit toy example over that yields a MDS code and demonstrates how to realize AG-Goppa codes by selecting appropriate and a corresponding curve equation. Overall, the work offers a practical, data-efficient method to obtain Riemann–Roch bases and to deploy hyperelliptic-curve codes in coding applications.

Abstract

For an (imaginary) hyperelliptic curve of genus , with a Weierstrass point , taken as the point at infinity, we determine a basis of the Riemann-Roch space , where is of degree zero, directly from the Mumford representation of . This provides in turn a generating matrix of a Goppa code.
Paper Structure (3 sections, 2 theorems, 37 equations)

This paper contains 3 sections, 2 theorems, 37 equations.

Table of Contents

  1. Notations and reduction to the Mumford representation
  2. Main theorem
  3. Applications to coding theory

Key Result

Theorem 1

Given the hyperelliptic curve $\mathcal{H}$ of genus $g$ and degree $d=2g+1$ defined by H, given the divisor $D=\Delta+m \Omega$ of positive degree $m$ on $\mathcal{H}$ defined in Remark auxiliar, with $\Delta=\operatorname{div}(u(x),v(x))$ in Mumford representation, let $t:=\operatorname{deg} u(x)\ for $\operatorname{char}\mathsf{k}=p>2$, and $\Psi(x,y)=\frac{y+ v(x) + h(x)}{u(x)}$, for $p=2$. If

Theorems & Definitions (9)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Theorem 2
  • proof
  • Remark 4
  • Example 1