AG码的等距双标志
Isometry-Dual Flags of AG Codes
摘要：
考虑有限域$\mathbb{F}$上长度为$n$的单点AG码的完整标志$\{0\} = C_0 < C_1 < \cdots < C_n = \mathbb{F}^n$。通过在不同于$Q$的点$P_1,\dots,P_n$中对在给定点$Q$处具有极点的函数求值来定义代码。如果给定的标志和对应的对偶标志在等距上相同，则标志具有等距对偶属性。对于八元域上的射影直线、Hermitian曲线、Suzuki曲线、Ree曲线和Klein曲线等曲线，在不同于点$Q$的所有有理点上求值得到的极大标志是自对偶的.更一般地说，我们问是否通过评估在一个适当的子集的合理点的标志是等距对偶。文献[3]证明了，对于亏格为$g$的曲线，用$n > 2g+2$有理点子集定义的单点AG码的标志是等距对偶的当且仅当标志中的最后一个码$C_n$是用极点阶至多为$n+2g-1$的函数定义的。使用不同的方法，我们将此特征扩展到大小为$n \geq 2g+2$的所有子集。此外，我们表明，这是最好的可能性，通过给出的例子，等距双标志与$n=2g+1$，使$C_n$是由最多$n+2g-2$的极点阶的功能。我们还证明了一个必要条件，制定的最大稀疏理想的Weierstrass半群的$Q$，在该标志下的打孔单点AG码继承的等距对偶属性从原来的unpunctured标志。
英文摘要：
Consider a complete flag $\{0\} = C_0 < C_1 < \cdots < C_n = \mathbb{F}^n$ of one-point AG codes of length $n$ over the finite field $\mathbb{F}$. The codes are defined by evaluating functions with poles at a given point $Q$ in points $P_1,\dots,P_n$ distinct from $Q$. A flag has the isometry-dual property if the given flag and the corresponding dual flag are the same up to isometry. For several curves, including the projective line, Hermitian curves, Suzuki curves, Ree curves, and the Klein curve over the field of eight elements, the maximal flag, obtained by evaluation in all rational points different from the point $Q$, is self-dual. More generally, we ask whether a flag obtained by evaluation in a proper subset of rational points is isometry-dual. In [3] it is shown, for a curve of genus $g$, that a flag of one-point AG codes defined with a subset of $n > 2g+2$ rational points is isometry-dual if and only if the last code $C_n$ in the flag is defined with functions of pole order at most $n+2g-1$. Using a different approach, we extend this characterization to all subsets of size $n \geq 2g+2$. Moreover we show that this is best possible by giving examples of isometry-dual flags with $n=2g+1$ such that $C_n$ is generated by functions of pole order at most $n+2g-2$. We also prove a necessary condition, formulated in terms of maximum sparse ideals of the Weierstrass semigroup of $Q$, under which a flag of punctured one-point AG codes inherits the isometry-dual property from the original unpunctured flag.
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