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Session S29 - Theory and Applications of Coding Theory

Monday, July 19, 18:00 ~ 18:25 UTC-3

Isometry-Dual Property in Flags of Two-Point AG codes

Alonso Sepúlveda Castellanos

University Federal of Uberlândia, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

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 $supp(D)$. 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 $y^m=f(x)$, 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 $f(x)$, 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.

Joint work with Luciane Quoos (University Federal of Rio de Janeiro, Brazil) and Maria Bras-Amorós (Universitat Rovira i Virgili, Spain).

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