Let ${\mathbb F}_q$ be the finite field with $q$ elements and let ${\mathbb N}$ be the set of non-negative integers. A flag of linear codes $C_0 \subsetneq C_1 \subsetneq \cdots \subsetneq C_s$ is said to have the {\it isometry-dual property} if there exists a vector ${\bf x}\in (\mathbb{F}_q^*)^n$ such that $C_i={\bf x} \cdot C_{s-i}^\perp$, where $C_i^\perp$ denotes the dual code of the code $C_i$. Consider ${\mathcal F}$ a function field over ${\mathbb F}_q$, and let $P$ and $Q_1,\ldots, Q_t$ be rational places in ${\mathcal F}$. Let the divisor $D$ be the sum of pairwise different places of ${\mathcal F}$ such that $P, Q_1,\ldots, Q_t$ are not in $\mbox{supp}(D)$, and let ${\bf G}_{\boldsymbol\beta}$ be the divisor $\sum_{i=1}^t\beta_iQ_i$, for given $\beta_i's \in {\mathbb Z}$. For suitable values of $\beta_i's$ in ${\mathbb Z}$ and varying an integer $a$ we investigate the existence of isometry-dual flags of codes in the families of many-point algebraic geometry codes $$C_\mathcal L(D, a_0P+{\bf G}_{\boldsymbol\beta})\subsetneq C_\mathcal L(D, a_1P+{\bf G}_{\boldsymbol\beta}))\subsetneq \dots \subsetneq C_\mathcal L(D, a_sP+{\bf G}_{\boldsymbol\beta})).$$ We then apply the obtained results to the broad class of Kummer extensions ${\mathcal F}$ defined by affine equations of the form $y^m=f(x)$, for $f(x)$ a separable polynomial of degree $r$, where $\gcd(r, m)=1$. In particular, depending on the place $P$, we obtain necessary and sufficient conditions depending on $m$ and $\beta_i$'s such that the flag has the isometry-dual property.

中文翻译：

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多点代数几何代码标志中的等距对偶性质

令 ${\mathbb F}_q$ 是具有 $q$ 元素的有限域，令 ${\mathbb N}$ 是非负整数的集合。如果存在向量 ${\bf x}\in (\mathbb{F} _q^*)^n$ 使得 $C_i={\bf x} \cdot C_{si}^\perp$，其中 $C_i^\perp$ 表示代码 $C_i$ 的对偶代码。考虑 ${\mathcal F}$ 是 ${\mathbb F}_q$ 上的一个函数域，并让 $P$ 和 $Q_1,\ldots, Q_t$ 是 ${\mathcal F}$ 中的有理位。令除数 $D$ 为 ${\mathcal F}$ 的成对不同位置之和，使得 $P, Q_1,\ldots, Q_t$ 不在 $\mbox{supp}(D)$ 中，并令 $ {\bf G}_{\boldsymbol\beta}$ 是除数 $\sum_{i=1}^t\beta_iQ_i$，对于给定的 $\beta_i 的 \in {\mathbb Z}$。对于 $\beta_i' 的合适值 s$ in ${\mathbb Z}$ 和改变整数 $a$ 我们研究了多点代数几何代码族中代码的等距对偶标志 $$C_\mathcal L(D, a_0P+{\ bf G}_{\boldsymbol\beta})\subsetneq C_\mathcal L(D, a_1P+{\bf G}_{\boldsymbol\beta}))\subsetneq \dots \subsetneq C_\mathcal L(D, a_sP+{ \bf G}_{\boldsymbol\beta})).$$ 然后我们将获得的结果应用到由 $y^m=f( x)$，对于$f(x)$ 为$r$ 次的可分离多项式，其中$\gcd(r, m)=1$。特别是，根据位置 $P$，我们根据 $m$ 和 $\beta_i$ 获得充分必要条件，使得标志具有等距对偶性质。