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

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