发表机构
Universidade Federal do Rio de Janeiro; Universidade Estadual de Campinas (UNICAMP)（里约热内卢联邦大学; 坎皮纳斯州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究超椭圆曲线上代数几何码，通过芒福德表示刻画非特殊除子，构造LCP码并给出MDS准则，在特定条件下得到LCD码，应用于极大超椭圆曲线给出特定参数的MDS LCD码示例并推测其对所有$q \geq 4$存在。
AI 中文摘要
我们研究亏格$g \geq 2$且具有互补性质的超椭圆曲线上的代数几何码。首先，通过约化芒福德表示的多项式次数来刻画次数为$g$和$g - 1$的非特殊除子，将经典几何难题转化为单变量多项式的单次数测试。利用此方法，通过雅可比上的多项式运算构造码的线性互补对（LCP），并给出所得码为最大距离可分（MDS）码的芒福德次数准则。在雅可比的2 - 挠条件下，得到明确的乘数使这些对成为线性互补对偶（LCD）码。最后，将此框架应用于$\mathbb{F}_{q^2}$上的极大超椭圆曲线$\mathcal{X} \colon y^2 = x^q + x$，给出参数为$[2q,q,q + 1]_{q^2}$的MDS LCD码的显式示例，$q = 4,5,7$经计算验证；我们推测在启发式支持下，对于所有$q \geq 4$此类码都存在。
英文摘要
We study algebraic geometry codes on hyperelliptic curves of genus $g \geq 2$ 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 $q\ge5$ the existence of MDS LCD codes with parameters $[2q,q,q+1]_{q^2}$ arising from this construction.
Comments20 pages
