2026/07/18 by Adler Marques, Yuri da Silva, Saeed Tafazolian
#math.AG #cs.IT #math.IT
We study algebraic geometry codes on hyperelliptic curves of genus g ≥ 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 X \colon y2 = xq + x over \mathbbFq2 and give explicit examples of MDS LCD codes with parameters [2q,q,q+1]q2 for q = 4, 5, 7, verified computationally; we conjecture, with heuristic support, that such codes exist for all q ≥ 4.