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

Abstract: A flag of codes $$C_0 \subsetneq C_1 \subsetneq \cdots \subsetneq C_s \subseteq {\mathbb F}<em>q<sup>n$$ is said to satisfy the {\it isometry-dual property} if there exists ${\bf x}\in (\mathbb{F}_q<sup>*)<sup>n$ such that the code $C_i$ is {\bf x}-isometric to the dual code $C</em>{s-i}<sup>\perp$ 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 $$C_\mathcal L(D, a_0P+bQ)\subsetneq C_\mathcal L(D, a_1P+bQ)\subsetneq \dots \subsetneq C_\mathcal L(D, a_sP+bQ),$$ where the divisor $D$ is the sum of pairwise different rational places of ${\mathcal F}$ and $P, Q$ are not in $\mbox{supp}(D)$. 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 $y<sup>m=f(x)$, for $f(x)$ a separable polynomial of degree $r$, where $\mbox{gcd}(r, m)=1$. For $P$ the rational place at infinity and $Q$ the rational place associated to one of the roots of $f(x)$, 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.