The Isometry-Dual Property in Flags of Many-Point Algebraic Geometry Codes

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.