标题： 多处纯隙和任意Kummer扩张的多点AG码 显示英文标题

标题： Pure Gaps at Many Places and Multi-point AG Codes from Arbitrary Kummer Extensions

摘要： 对于由仿射方程 $y^{m}=\prod_{i=1}^{r} (x-\a_i)^{\lambda_i}$ 定义的，且定义在有限域 $\fq$ 的代数扩张 $K$ 上的库默尔扩域，其中 $\la_i\in \Z\backslash\{0\}$ 对于 $1\leq i\leq r$, $\gcd(m,q) = 1$, 和 $\a_1,\cdots,\a_r\in K$ 是两两不同的元素，我们提出了一种简单而高效的方法来寻找许多完全分歧点的所有纯间隙。 我们引入了纯间隙的下界集合，并指出纯间隙集完全由下界集合确定。 此外，我们证明了一个纯间隙可以通过轻松验证仅一个不等式从已知的纯间隙推导出来。 然后，在 $\lambda_1 = \lambda_2 = \cdots = \lambda_r$ 成立的情况下，我们完全确定了多个完全分歧点处纯间隙集的显式描述， 这包括这些点的集合包含无穷远点的情形。 最后，我们将这些结果应用于构造具有优良参数的多点代数几何码。 作为一个例子，参数为 $[74, 60, \geq 10]$ 的 $\mathbb{F}_{25}$ 上的码字创造了新的记录。 显示英文摘要

摘要： For a Kummer extension defined by the affine equation $y^{m}=\prod_{i=1}^{r} (x-\a_i)^{\lambda_i}$ over an algebraic extension $K$ of a finite field $\fq$, where $\la_i\in \Z\backslash\{0\}$ for $1\leq i\leq r$, $\gcd(m,q) = 1$, and $\a_1,\cdots,\a_r\in K$ are pairwise distinct elements, we propose a simple and efficient method to find all pure gaps at many totally ramified places. We introduce a bottom set of pure gaps and indicate that the set of pure gaps is completely determined by the bottom set. Furthermore, we demonstrate that a pure gap can be deduced from a known pure gap by easily verifying only one inequality. Then, in the case where $\lambda_1 = \lambda_2 = \cdots = \lambda_r$, we fully determine an explicit description of the set of pure gaps at many totally ramified places, This includes the scenario in which the set of these places contains the infinite place. Finally, we apply these results to construct multi-point algebraic geometry codes with good parameters. As one of the examples, a presented code with parameters $[74, 60, \geq 10]$ over $\mathbb{F}_{25}$ yields a new record.