Explicit LCP of MDS Codes and LCD Codes on Hyperelliptic Curves via Mumford Representation

2026-07-18

AI Summary

This work addresses the efficient construction of algebraic geometry codes that simultaneously satisfy the maximum distance separable (MDS) and linear complementary dual (LCD) properties. By leveraging the reduced Mumford representation on the hyperelliptic curve $y^2 = x^q + x$, the authors transform the nonspeciality criterion for divisors into a univariate polynomial degree test, yielding a Mumford-degree-based MDS criterion. Combining this with a 2-torsion condition in the Jacobian, they explicitly construct linear complementary pair (LCP) codes. The method successfully produces MDS LCD codes over $\mathbb{F}_{q^2}$ with parameters $[2q, q, q+1]$, verified for $q = 4, 5, 7$, and conjectured to hold for all $q \geq 4$. This significantly advances the explicit, structured construction of efficient MDS LCD codes.

Abstract

We study algebraic geometry codes on hyperelliptic curves of genus $g \geq 2$ with complementarity properties. Our first contribution is a characterization of non-special divisors of degree $g$ and $g-1$ via the polynomial degrees of their reduced Mumford representation, reducing a classical hard geometric problem to a single-degree test on univariate polynomials. Using this, we construct Linear Complementary Pairs (LCP) of codes via polynomial arithmetic on the Jacobian and provide a criterion in terms of Mumford degrees for the resulting codes to be Maximum Distance Separable (MDS). Under a $2$-torsion condition in the Jacobian, equivalently a divisibility condition on the Mumford polynomials, we obtain explicit multipliers that turn these pairs into Linear Complementary Dual (LCD) codes. Finally, we apply this framework to the maximal hyperelliptic curve $\mathcal{X} \colon y^2 = x^q + x$ over $\mathbb{F}_{q^2}$ and give explicit examples of MDS LCD codes with parameters $[2q,q,q+1]_{q^2}$ for $q = 4, 5, 7$, verified computationally; we conjecture, with heuristic support, that such codes exist for all $q \geq 4$.

Problem

• LCP codes
• LCD codes
• MDS codes
• hyperelliptic curves
• Mumford representation

Innovation

• Mumford representation
• Linear Complementary Dual codes
• MDS codes
• hyperelliptic curves
• algebraic geometry codes