Pure Gaps at Many Places and Multi-point AG Codes from Arbitrary Kummer Extensions
Huachao Zhang, Chang-An Zhao
TL;DR
This work advances the construction of multi-point algebraic geometry codes by developing a bottom-set framework for pure gaps on Kummer extensions defined by $y^{m}=\prod_{i=1}^{r}(x-\alpha_i)^{\lambda_i}$. The authors prove that the full set of pure gaps at many totally ramified places is determined by a bottom set and provide an efficient inductive procedure to enumerate them, including a one-inequality rule to generate new gaps from known ones. They specialize to special Kummer extensions, giving explicit union-based descriptions of bottom-gap sets in several regimes, and derive numerous families of consecutive pure gaps. These results yield practical code constructions, including multiple multi-point differential AG codes with parameters surpassing existing records, notably a $[74,60,\ge 10]$ code over $\mathbb{F}_{25}$. The methodology broadens explicit gap descriptions for arbitrary multiplicities and enhances the design of high-distance, multi-point AG codes with concrete, verifiable criteria.
Abstract
For a Kummer extension defined by the affine equation $y^{m}=\prod_{i=1}^{r} (x-\a_i)^{λ_i}$ 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 $λ_1 = λ_2 = \cdots = λ_r$, 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.