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

A flag of codes $C_{0} \subsetneq C_{1} \subsetneq \cdots \subsetneq C_{s} \subseteq \mathbb {F}_{q} ^{n}$ is said to satisfy the isometry-dual property if there exists ${\mathbf{x}}\in (\mathbb {F}_{q}^{*})^{n}$ such that the code $C_{i}$ is x-isometric to the dual code $C_{s-i}^\perp $ for all $i=0,\ldots, s$ . For $P$ and $Q$ rational places in a function field $\mathcal {F}$ , we investigate the existence of isometry-dual flags of codes in the families of two-point algebraic geometry codes $C_{\mathcal {L}}(D, a_{0}P+bQ)\subsetneq C_{\mathcal {L}}(D, a_{1}P+bQ)\subsetneq {\dots } \subsetneq C_{\mathcal {L}}(D, a_{s}P+bQ)$ , where the divisor $D$ is the sum of pairwise different rational places of $\mathcal {F}$ and $P, Q$ are not in $\mathop {\mathrm {supp}}\nolimits (D)$ . We characterize those sequences in terms of $b$ for general function fields. We then apply the result 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 $\gcd (r, m)=1$ . For $P$ the rational place at infinity and $Q$ the rational place associated to one of the roots of $f(x)$ , and for $D$ an $Aut(\mathcal {F}/ \mathbb {F}_{q})$ -invariant sum of rational places of $\mathcal {F}$ , such that $P, Q \notin \mathop {\mathrm {supp}}\nolimits D$ , it is shown that the flag of two-point algebraic geometry codes has the isometry-dual property if and only if $m$ divides $2b+1$ . At the end we illustrate our results by applying them to two-point codes over several well know function fields.