Isometry-Dual Flags of Many-Point AG Codes

TL;DR

This paper investigates the isometry-dual property of flags of algebraic geometry codes from function fields, extending previous results to multi-point codes with negative integers and applying findings to Kummer extensions.

Contribution

It extends the characterization of isometry-dual flags to multi-point codes with negative parameters and applies these results to Kummer extensions, broadening the class of codes analyzed.

Findings

Extended results to negative integers in code parameters.

Derived necessary and sufficient conditions for isometry-dual property in Kummer extensions.

Provided criteria for flags of multi-point codes to satisfy the isometry-dual property.

Abstract

Let $F_q$ be a finite field. A flag of $F_q$-linear codes $C_0\subsetneq C_1\subsetneq\dots\subsetneq C_s$ is said to satisfy the isometry-dual property if there exists a vector $x\in(F_q^*)^n$ such that $C_i=x\cdot C_{s-i}^\perp$, where $C_i^\perp$ denotes the dual code of $C_i$. Consider $F/F_q$ a function field and let $P$ and $Q_1,\ldots,Q_t$ be rational places of $F$. Let the divisor $D$ be the sum of pairwise different places of $F$ such that $P, Q_1,\dots,Q_t$ are not in $supp(D)$. In a previous work we investigated the existence of flags of two-point codes $C(D,a_0P+bQ_1)\subsetneq C(D,a_1P+bQ_1))\subsetneq\dots\subsetneq C(D,a_sP+bQ_1)$ satisfying the isometry-dual property for a non-negative integer $b$ and an increasing sequence of positive integers $a_0,\dots,a_s$. While for one-point codes (i.e. for $b=0$) there is only need to analyze positive integers $a$, for the case of $(t+1)$-point codes, the integers $a$ may be negative. We extend our previous results in different directions. On one hand to the case of negative integers $a$ and $b$, and on the other hand we extend our results to flags of $(t+1)$-point codes $C(D,a_0P+\sum_{i=1}^t\beta_iQ_i)\subsetneq C(D, a_1P+\sum_{i=1}^t\beta_iQ_i))\subsetneq\dots\subsetneq C(D, a_sP+\sum_{i=1}^t\beta_iQ_i)$ for any tuple of (either positive or negative) integers $\beta_1,\dots,\beta_t$ and for an increasing sequence of (either positive or negative) integers $a_0,\dots,a_s$. We apply the obtained results to the broad class of Kummer extensions defined by affine equations of the form $y^m=f(x)$, for $f(x)$ a separable polynomial of degree $r$, where $gcd(r, m)=1$. In particular, depending on the place $P$ and for $D$ an $Aut(F_q(x, y)/F_q(x))$-invariant sum of rational places of $F$ such that $P,Q_i\notin supp(D)$, we obtain necessary and sufficient conditions on $m$ and $\beta_i$'s such that the flag has the isometry-dual property.