Explicit LCP of MDS Codes and LCD Codes on Hyperelliptic Curves via Mumford Representation
Abstract
We study algebraic geometry codes on hyperelliptic curves of genus g ≥ 2 equipped with a rational Weierstrass point, with a focus on 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 geometric problem to a 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. We give an MDS LCP with parameters [8,4,5]16 in characteristic two, an MDS LCD code with parameters [10,5,6]25 on the maximal hyperelliptic curve y2=x5+x, and an LCD code with parameters [28,14]49 on y2=x7+x. Motivated by the explicit q=5 computation, we conjecture for odd prime powers q≥5 the existence of MDS LCD codes with parameters [2q,q,q+1]q2 arising from this construction.
Related
- MDS Algebraic Geometry Codes of Square-Root Length on Hyperelliptic Curves
- A cubic Chabauty criterion for hyperelliptic curves
- Normal Matrix Model and Hyperelliptic Curves
- Maximal singular curves over finite fields from elliptic and hyperelliptic curves
- Arithmetic statistics of isogeny Selmer groups associated to hyperelliptic curves
BibTeX
@misc{marques2026explicit,
title = {{Explicit LCP of MDS Codes and LCD Codes on Hyperelliptic Curves via Mumford Representation}},
author = {Adler Marques and Yuri da Silva and Saeed Tafazolian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.16945},
eprint = {2607.16945},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.16945},
}
RIS
TY - UNPB
TI - Explicit LCP of MDS Codes and LCD Codes on Hyperelliptic Curves via Mumford Representation
AU - Adler Marques
AU - Yuri da Silva
AU - Saeed Tafazolian
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2607.16945
UR - https://arxiv.org/abs/2607.16945
AB - We study algebraic geometry codes on hyperelliptic curves of genus g ≥ 2 equipped with a rational Weierstrass point, with a focus on 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 geometric problem to a 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. We give an MDS LCP with parameters [8,4,5]_16 in characteristic two, an MDS LCD code with parameters [10,5,6]_25 on the maximal hyperelliptic curve y^2=x^5+x, and an LCD code with parameters [28,14]_49 on y^2=x^7+x. Motivated by the explicit q=5 computation, we conjecture for odd prime powers q≥5 the existence of MDS LCD codes with parameters [2q,q,q+1]_q^2 arising from this construction.
ER -