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

TL;DR

This paper investigates the isometry-dual property in flags of two-point algebraic geometry codes, characterizing when such flags are dual-isometric in terms of parameters in function fields and applying results to Kummer extensions.

Contribution

It provides a characterization of the isometry-dual property for two-point algebraic geometry codes in terms of divisors and applies this to Kummer extensions, offering explicit conditions.

Findings

The isometry-dual property depends on the parameter b in the divisor for general function fields.

For Kummer extensions, the property holds if and only if m divides 2b+1.

Results are illustrated with examples over well-known function fields.

Abstract

A flag of codes $C_0 \subsetneq C_1 \subsetneq \cdots \subsetneq C_s \subseteq {\mathbb F}_q^n$ is said to satisfy the {\it isometry-dual property} if there exists ${\bf x}\in (\mathbb{F}_q^*)^n$ such that the code $C_i$ is {\bf x}-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 $$C_\mathcal L(D, a_0P+bQ)\subsetneq C_\mathcal L(D, a_1P+bQ)\subsetneq \dots \subsetneq C_\mathcal L(D, a_sP+bQ),$$ 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 $y^m=f(x)$, 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.