Explicit LCP of MDS Codes and LCD Codes on Hyperelliptic Curves via Mumford Representation
Information Theory
Summary
The authors investigate special error-correcting codes that come from certain algebraic curves called hyperelliptic curves with genus at least 2. They find a simpler way to identify important building blocks for these codes using degrees of certain polynomials, making a complex geometric problem easier. Using this insight, they build pairs of codes with complementary properties and describe when these codes reach the best possible distance given their length and size. They also show how to transform these pairs into codes with linear complementary duality under special conditions and provide concrete examples using a well-known hyperelliptic curve over finite fields. The authors conjecture their construction works broadly for larger parameters.
Authors
Adler Marques, Yuri da Silva, Saeed Tafazolian
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 $q \geq 4$.