Isometry-Dual Flags of Many-Point AG Codes

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 $C(D,a_0P+\sum_{i=1}^tβ_iQ_i)\subsetneq C(D, a_1P+\sum_{i=1}^tβ_iQ_i))\subsetneq\dots\subsetneq C(D, a_sP+\sum_{i=1}^tβ_iQ_i)$ for any tuple of (either positive or negative) integers $β_1,\dots,β_t$ 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 $β_i$'s such that the flag has the isometry-dual property.