The Isometry-Dual Property in Flags of Two-Point Algebraic Geometry Codes
Maria Bras-Amorós, Alonso S. Castellanos, Luciane Quoos · IEEE Transactions on Information Theory · 2021
A flag of codes$C_{0} \subsetneq C_{1} \subsetneq \cdots \subsetneq C_{s} \subseteq \mathbb {F}_{q} ^{n}$is said to satisfy theisometry-dual propertyif there exists${\mathbf{x}}\in (\mathbb {F}_{q}^{*})^{n}$such that the code$C_{i}$isx-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}} olimits (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 otin \mathop {\mathrm {supp}} olimits 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.