# Isometry-Dual Flags of AG Codes

## TL;DR

This paper investigates the conditions under which flags of algebraic geometry codes, derived from evaluations at rational points on algebraic curves, exhibit the isometry-dual property, extending known results and providing new examples and necessary conditions.

## Contribution

It extends the characterization of isometry-dual flags to all subsets of rational points of size at least 2g+2 and provides examples and necessary conditions related to Weierstrass semigroups.

## Key findings

- Flags with n ≥ 2g+2 are isometry-dual iff the last code is defined with functions of pole order ≤ n+2g-1.
- Examples of isometry-dual flags with n=2g+1 are constructed, with pole order ≤ n+2g-2.
- A necessary condition involving maximum sparse ideals of the Weierstrass semigroup is established.

## Abstract

Consider a complete flag $\{0\} = C_0 < C_1 < \cdots < C_n = \mathbb{F}^n$ of one-point AG codes of length $n$ over the finite field $\mathbb{F}$. The codes are defined by evaluating functions with poles at a given point $Q$ in points $P_1,\dots,P_n$ distinct from $Q$. A flag has the isometry-dual property if the given flag and the corresponding dual flag are the same up to isometry. For several curves, including the projective line, Hermitian curves, Suzuki curves, Ree curves, and the Klein curve over the field of eight elements, the maximal flag, obtained by evaluation in all rational points different from the point $Q$, is self-dual. More generally, we ask whether a flag obtained by evaluation in a proper subset of rational points is isometry-dual. In [3] it is shown, for a curve of genus $g$, that a flag of one-point AG codes defined with a subset of $n > 2g+2$ rational points is isometry-dual if and only if the last code $C_n$ in the flag is defined with functions of pole order at most $n+2g-1$. Using a different approach, we extend this characterization to all subsets of size $n \geq 2g+2$. Moreover we show that this is best possible by giving examples of isometry-dual flags with $n=2g+1$ such that $C_n$ is generated by functions of pole order at most $n+2g-2$. We also prove a necessary condition, formulated in terms of maximum sparse ideals of the Weierstrass semigroup of $Q$, under which a flag of punctured one-point AG codes inherits the isometry-dual property from the original unpunctured flag.